Skip to main content

Full text of "Readable Relativity"

See other formats

FELfflWlY \ 


■ ■ ■ ■ ■■■■ ■ ■■■■■■■ 




TH bkkP 



■SI'-:'. Mi 







When the ideas involved in Einstein's work have 
become familiar, as they will do when they are 
taught in schools, Certain changes in our habits 
of thought are hkely to result, and to have great 
importance in the long run." 

Berth and Russell, 

The ABC of Relativity. 








Relativity without mathematics may be compared with 
" Painless Dentistry," or " Ski-ing without falling," or 
" Reading without Tears." Its ideas have, of course, been 
sketched in popular style by many writers, but precision 
can only be achieved by setting out the arguments in a 
mathematical form, and this precision is essential for a firm 
grasp of the fundamental principles of the subject. This 
book attempts to secure as high a degree of definition as is 
compatible with the standard of mathematical knowledge 
of the average person. The limitations this imposes are 
obvious, but inevitable if the subject is to lie within the 
sphere of general education. Given, however, this small 
amount of mathematical capacity and preferably also a 
willingness to work out a few numerical examples to test 
appreciation of the ideas peculiar to the subject, it should 
be possible to make Einstein's view of the Universe as 
much a part of the intellectual equipment of ordinary 
people as is that of Newton. 

It is difficult to specify all the books from which I have 
received assistance, but mention must be made of Pro- 
fessor Nunn's Relativity and Gravitation, and Professor 
Eddington's Space, Time, and Gravitation and Mathematical 
Theory of Relativity ; I am also indebted to Mr. Greenstreet 
for allowing me to make use of the articles in the Mathe- 
matical Gazette to which reference is made in Chapter II, 


and to Professor Eddington for information about the solar 
eclipse expedition of 1922. The subject-matter of the brief 
note on Mr. Adam's work on the spectrum of the companion 
star of Sirius is taken from Mr. Adam's published paper on 
the subject, which Professor Turner very kindly showed me. 
I have also to thank Mr. W. L. F. Browne for reading the 
text and making a number of valuable suggestions. 

C. V. D. 

February 1926. 



I. The Progress of Science. 
II. Alice through the Looking-Glass 

III. The Velocity of Light 

IV. Clocks . 
V. Algebraic Relations 

VI. Separation of Events 
VII. The Fourth Dimension 
VIII. Mass and Momentum 
IX. General Relativity 
X. The Einstein Tests 










" We know very little and yet it is astonishing that we know 
so much, and still more astonishing that so little knowledge 
can give us so much power." — Bertrand Russell, The 
ABC of Relativity. 

Our Outdoor World. 

Every one nowadays has heard of Einstein and Relativity. 
The man in the street, however, still treats the subject 
more as a fairy tale or a mathematical jest than as a 
contribution to scientific knowledge and method which is 
as momentous (if it stands th« test of continued scrutiny) 
as any that has ever been made. Relativity is a branch of 
physics, not of pure mathematics. Its conclusions could 
not, of course, have been obtained without the aid of pure 
mathematical reasoning of a difficult and abstruse nature, 
but the mathematical side is incidental : mathematics has 
merely supplied the machinery for working on the material 
available and the language for describing the result. The 
material itself i& the product of experiment, observation, and 
measurement. The whole aim of Natural Science is to 
examine what is happening in our outdoor world, the 
Universe in which we live, and to construct the simplest 
set of suppositions which will cover all the observed facts. 

The sole question, therefore, that arises is this : Does 


Einstein's theory of Relativity give a more harmonious 
and adequate picture of what is observed to be happening 
in the Universe than any other alternative scientific theory 
dealing with the same data, or not ? 

Many articles in newspapers and magazines seem to 
suggest that the Relativity creed contains such fantastic 
propositions that, however much they may appeal to 
philosophers, they cannot be taken seriously by the matter- 
of-fact individual who prides himself on believing only 
what he sees and distrusts suggestions which seem to 
contradict his own personal experience. It is therefore 
essential to realise that Einstein's theory stands or falls by 
observed facts : it aims at describing how things such as 
matter, time, and space do really behave. If, and when, 
any one can offer a simpler and more comprehensive account 
of the outdoor world than Einstein's theory is able to do, 
then Relativity will be superseded or modified. But at the 
present time it is contended that no other picture of the 
Universe, as adequate as Einstein's, exists. The story of 
scientific progress illustrates how indigestible new ideas are 
as a diet for plain, blunt men, but equally that those ideas 
which stand the test of time are easily assimilated by the 
plain, blunt men of a later date. 

The Ptolemaic System. 

Tradition suggests that Pythagoras (550 B.C.) was the 
first to teach that the Earth is a sphere, poised in space. 
This puzzled the man in the street in two respects : (i) How 
was the Earth supported ? (ii) Why did not people or 
things on the other side of the Earth, being upside down, 
fall off ? 

Here, then, was one of the first of a series of shocks 
scientists have administered to common sense and will 
doubtless continue to administer as long as knowledge 
increases. But many centuries passed before the sphericity 


of the Earth was generally accepted, at any rate outside 
Greece ; in fact it needed the circumnavigation of the 
world to drive it home as a matter of practical experience. 
Even to-day there are some who still maintain the Earth 
is flat (cf. Kipling's story, " The Village that voted the 
Earth was Flat," in A Diversity of Creatures), just as there 
are others who still try to square the circle. The Greek 
scientists, however, waived criticism aside and proceeded 
to calculate the circumference and diameter of the Earth. 
Aristotle (350 B.C.) states that the mathematicians of his 
time made the circumference 400,000 stadia (probably 
about 40,000 miles), but a far more accurate result was 
obtained by Eratosthenes (250 B.C.), who gave the circum- 
ference as the equivalent of 24,700 miles and the radius 
as 3925 miles, a degree of accuracy far higher than the 
rough methods he employed justify ; it was in fact a 
lucky result. A later but less accurate measurement was 
made by Ptolemy (140 a.d.), whose treatise, largely astro- 
nomical, The Almagest, dominated scientific thought for 
the next fourteen centuries. The Ptolemaic System placed 
the Earth at the centre of the Universe, and regarded the 
Sun and the planets as moving round it in paths built up 
of circle and epicycles (i.e. circles rolling on circles) ; and 
although as time passed minor modifications were made 
to bring the theory into closer accord with observation, 
the general principle was accepted without question, till 
the time of Copernicus. It may be of interest to record 
that a contrary view had been put forward before Ptolemy's 
time. A certain Aristarchus of Samos (310-230 B.C.) 
maintained that the Sun was the centre of the world and 
that both the planets and the Earth moved round it. But 
this was far too advanced a doctrine not only for the 
common man but also for the scientists of his day ; it 
offended their sense of propriety in that it degraded the 
Earth from the central to a subordinate position in Nature, 


and it outraged common sense to suppose that the Earth, 
which, as they said, any one could see and feel to be at 
rest, was really travelling through space more than a 
million miles a day. [Eratosthenes estimated the distance 
of the sun as 80,000,000 miles.] We are all now so accus- 
tomed to think of the Sun as the centre of the solar 
system that it is difficult to realise what a shock the 
normal man received when the doctrine was first seriously 

The Copernican System. 

The theory of Copernicus, De Revolutionibus orbium 
Coelestium, was published in 1543. It regarded the Sun as 
at rest at the centre of the world with the Earth and the 
other planets moving round it in circular orbits. The 
laborious work of Kepler (1571-1630), although showing 
that the orbits were not circular, gave powerful support 
to the heliocentric principle of Copernicus by establishing 
that the planets could be regarded as moving in ellipses 
with the Sun at one focus, and that the sizes of the orbits 
and the times and rates of description obeyed two simple 
quantitative laws, which later on assisted Newton to 
formulate his law of Universal Gravitation. The invention 
of the telescope in 1608 led Galileo (1564-1642) to discover 
Jupiter's moons, and this small-scale model of the solar 
system convinced him of the truth of the Copernican 
theory. After much hesitation and with considerable 
trepidation, which after-events fully justified, Galileo pub- 
lished in 1630 an account of his discoveries and beliefs. 
His thesis was not only a shock to the man in the street 
but, unhappily for Galileo, a shock also to the Church. 
In self-defence, he begged his opponents to come and look 
through his telescope, but neither the professors nor the 
ecclesiastics would do so. He was summoned to Rome, 
an old man in feeble health, tried by the Inquisition, and 


forced to " abjure, curse, and detest his errors and heresies." 
He died, a broken man and blind, near Florence in 1642. 
Einstein has good cause to be thankful that there is no 
Inquisition in power to-day. 

Universal Gravitation. 

The work of Copernicus, Kepler, Simon Marius, and 
Galileo was crowned by the publication of Newton's 
Principia in 1685. The notion that bodies fell to the earth 
owing to some form of attraction exerted by the earth did 
not originate with Newton. His genius showed itself in 
extending this idea to the whole Universe, formulating his 
result in a single law, and verifying it by an examination 
of the motion of the planets, comets, the Earth, and the 
Moon. Newton's troubles lay with his brother-scientists, 
not with ordinary folk, and many years of his life were 
embittered by the professional controversies which the 
Principia evoked. It has been said that there are com- 
paratively few scientists who can, at the present time, 
really understand the mathematical work which is the 
scaffolding of Einstein's theory, but there were relatively 
far fewer in Newton's day who could appreciate the 
reasoning of the Principia; and of course a long time 
elapsed before Newton's ideas became part of the equip- 
ment of the ordinary educated man, as they are to-day. 

The Mechanics of Galileo and Newton. 

It is necessary to state, however briefly, the fundamental 
principles upon which Newton, using the observations of 
Kepler and the ideas of Galileo, based his system of celestial 
mechanics. He followed Galileo in saying that the nature 
of a body is such that, if at rest it will remain at rest, and 
if in motion it will continue to move uniformly in a straight 
line, unless there is some external cause operating to pro- 
duce a change. This is the Principle of Inertia, and to 


the external cause he gives the name of Force and there- 
from develops his idea of Mass. 1 

For the working out of his mechanics, he postulates : 

(i) The notion of absolute time : time flows uniformly 
and without reference to anything else. 

(ii) The notion of absolute space : a fixed standard 
of reference, immutable and immovable, enabling the 
position or motion of any object in the Universe to be 
determined. The Earth is not at rest, the Sun may not 
be at rest, but there is, so Newton says, something existing 
in the Universe which will act as a fixed frame for defining 
absolute position and absolute motion. 

The normal man neither had nor has any difficulty in 
assenting to these suppositions ; in fact they seem so 
natural that he is shocked when asked to question them. 
The idea of universal gravitation was, on the other hand, 
far more perplexing. It appeared to involve the notion 
of " action at a distance," whereas everyday experience 
pointed to the belief that the action of one body on another 
was either caused by direct contact or some concrete 
connecting agency. Newton himself appears to have 
thought that further explanation was needed. To-day it 
is an idea that the normal man accepts without protest ; 
for him time and tradition have, as always, acted as the 
necessary shock-absorbers ; but none the less since Newton's 
time a succession of scientists has attempted by a variety 
of physical theories to bridge the gulf. 

Measuring Instruments. 

The human senses, unaided by mechanical assistance, 
are not adapted for making accurate measurements either 
of time or space. 

Time. — Our sense of duration for periods of any length 
depends mainly on whether our occupation is interesting 
1 Fawdry, Readable Mechanics, ch. vii. 


or tedious : our estimate of short periods of time is often 
ludicrously inaccurate. This may be illustrated by testing 
the ability of any one to judg'e the length of a minute, 
ruling out, of course, the assistance that counting, either 
aloud or mentally, affords. 

The use of the sun-dial, which treats the Sun as a clock, 
is known to date back to 1500 B.C. in Egypt, and according 
to tradition it was introduced into Greece from Babylon 
by Anaximander in the sixth century B.C. ; about this time 
also hour-glasses and water-clocks began to be made. Clocks 
depending on trains of wheels driven by a falling weight 
were used by the Romans in the sixth century a.d. The 
pendulum clock was invented by Huygens in 1673, ninety 
years after Galileo had discovered the isochronous property 
of a pendulum : Galileo himself used for his experimental 
work a form of water-clock l which gave surprisingly good 
results. The first ship's chronometer of a reliable nature 
was constructed by a Yorkshireman, John Harrison, in 
1761. This was an invention of immense practical import- 
ance, as at that time ships at sea could only determine 
their longitude by means of a clock. Under present condi- 
tions it is of less importance, because Greenwich mean 
time is sent out by wireless at noon each day. 

Space. — Any one who visits Oxford should go and see 
the Evans Collection of ancient scientific apparatus at the 
old Ashmolean. The delicacy and accuracy of workman- 
ship of the rulers, compasses, and astrolabes {i.e. instru- 
ments for measuring angles fixing the positions of the 
stars), some of which date back to very early times, can 
only be appreciated by seeing them. Chaucer wrote a 
treatise on the use of the astrolabe to make sure that his 
son should be properly instructed in its theory when he 
went to Oxford. The principle of the vernier, applied 
first to circular arcs, was discovered by a Portuguese 
1 Fawdry, Readable Mechanics, p. 62. 


named Nunes in 1542, and rediscovered by the Frenchman 
Vernier in 1631. 

Artificial Extension of the Senses. 

No great advance was possible in astronomy until the 
invention of the telescope in 1608. Naturally such an 
instrument at once aroused widespread attention, and 
within a few years telescopes were being used by scientists 
all over Europe. It not only increased the range of 
observation in astronomy, but it raised the degree of 
accuracy of measurement to a level far higher than had 
previously been possible. Modern discoveries are largely 
the fruit of two other inventions : (i) dry-plate photography, 
and especially its application to astronomy; (ii) the 
spectroscope and spectrum analysis. It is outside the 
province of this book to give any account of the amazing 
variety of application and the remarkable refinements 
of measurement yielded by these methods of research. 
The important point to observe is that the increase of 
knowledge of the Universe is due entirely to the successive 
aids that inventions have given in supplementing the 
powers of naked eyes and naked hands. Without such 
assistance, our knowledge of the structure of the world 
would remain very restricted. Without a telescope, we 
can only see details of things in our immediate neighbour- 
hood, and that only to a limited extent ; to fill in further 
detail, a microscope is necessary. Our ears only enable us 
to hear notes of limited pitch ; our eyes are sensitive to a 
range of colour far more limited than the range of the 
photographic plate ; if events happen too rapidly, our 
brains receive merely a blurred impression which it needs 
a slow-motion film to disentangle. 

Common Sense. 

Most of our outlook on life is coloured by the impressions 
we receive through senses, unaided by any artificial assist- 


ance. What we call a common-sense view of life is largely 
based on an acquaintance with things, confined in size 
between fairly narrow limits, restricted to small ranges 
of temperature and pressure, moving at low speeds and 
for short periods of time. It is not unreasonable to say 
that this gives us as true a view of the Universe as (say) 
a tourist could obtain of the interior of Westminster 
Abbey by looking through the keyhole of a side entrance. 
Successive inventions have enabled scientists to enlarge 
the keyhole, and perhaps at some future date will even 
throw open the door. If Science by its study of things 
very small and very large, very near and very distant, of 
temperatures very great, of velocities very high, is driven 
to conclusions which seem to violate our common-sense 
attitude, our keyhole notion of the Universe, it seems 
reasonable to treat it merely as one more shock in the 
succession which the man in the street has encountered 
and eventually absorbed. Nature is a conjurer for super- 
men. Generations of scientists have attempted to pene- 
trate her secrets. Bit by bit the disguise is being torn 
away, but each new discovery seems only to open out 
fresh avenues demanding further exploration. Nature is 
a true woman, who will have the last word. Scientists 
of every age may well echo Newton's account of his own 
life's work : " I do not know what I may appear to the 
world, but to myself I seem to have been only like a boy 
playing on the seashore, and diverting myself in now and 
then finding a smoother pebble or a prettier shell than 
ordinary, whilst the great ocean of truth lay all undis- 
covered before me." 


1. Why do you sometimes see on a cinematograph the wheels 
of a car rotating in a clockwise direction while the car 
itself is moving to the left ? 


2. A stone is tied to the end of a string and is whirled round 

in a circle horizontally. In what direction does the 
stone move when the string breaks ? 

3. If you assume that the earth describes a circle of radius 

93,000,000 miles in 365J days, how many miles does it 
move in a second ? 

4. Galileo found it easier to show that the acceleration of a 

falling body is constant by considering motion down a 
slope instead of vertically. How could a cinematograph 
operator improve on Galileo's treatment ? 

5. If you drop a stone and a feather at the same moment, do 

they hit the ground simultaneously ? Is your answer 
consistent with the statement that the acceleration due 
to gravity is the same for all bodies at the same place, 
regardless of their weights ? 

6. A camera photographs a 100 yards race for a cinema film, 

taking 150 exposures per second. They are reproduced 
on the screen at the rate of 15 per second ; how long, 
roughly, will the race appear to the audience to last ? 

7. Describe a vernier, and show how one can be made to read 

correct to yfoyth inch. 

8. Taking the length of the Equator as 25,000 miles, find the 

error in miles of the longitude of a position on the 
Equator, calculated from the record of a chronometer, 
if there is an error of one minute in the time. 

9. Eratosthenes found that the sun was in the zenith at Syene 

when it was 7 12' south of the zenith at Alexandria, 
which was known to be 5000 stadia north of Syene. 
What expression for the radius of the Earth can be 
deduced from these data ? 


" ' I can't believe that,' said Alice. 

" ' Can't you ? ' the Queen said in a pitying tone. ' Try 
again : draw a long breath and shut your eyes.' 

" Alice laughed : ' There's no use trying,' she said ; ' one 
can't believe impossible things.' 

" ' I daresay you haven't had much practice,' said the 
Queen. ' When I was younger, I always did it for half an 
hour a day. Why, sometimes I've believed as many as six im- 
possible things before breakfast.' " — Through the Looking-Glass . 

Can Nature deceive ? 

The scientists, in playing their game with Nature, are 
meeting an opponent on her own ground, who has not only 
made the rules of the game to suit herself, but may have 
even queered the pitch or cast a spell over the visiting team. 
If space possesses properties which distort our vision, 
deform our measuring-rods, and tamper with our clocks, is 
there any means of detecting the fact ? Can we feel hope- 
ful that "eventually cross-examination will break through 
the disguise ? Professor Garnett, 1 making use of Lewis 
Carroll's ideas, has given a most instructive illustration of 
a way in which Nature could mislead us, seemingly without 
any risk of exposure. 

Ultimately, we can only rely on the evidence of our 
senses, checked and clarified of course by artificial appara- 
tus, repeated experiment, and exhaustive inquiry. Observa- 
tions can often be interpreted unwisely, as an anecdote told 
by Sir George Greenhill illustrates : 

1 Mathematical Gazette, May 191 8. 


At the end of a session at the Engineering College, 
Coopers' Hill, a reception was held and the science depart- 
ments were on view. A young lady, entering the physical 
laboratory and seeing an inverted image of herself in a 
large concave mirror, naively remarked to her companion : 
" They have hung that looking-glass upside down." Had 
the lady advanced past the focus of the mirror, she would 
have seen that the workmen were not to blame. If 
Nature deceived her, it was at least a deception which 
further experiment would have unmasked. 

The Convex Looking-Glass. 

We shall now follow some of the adventures of Alice in 
a convex looking-glass world, as described by Professor 
Garnett. As a preliminary, it is necessary to enumerate 
some of the properties of reflection in a convex mirror. For 
the sake of any reader who wishes to see how they can be 
obtained, their proofs, which involve only the use of similar 
triangles and some elementary algebra, are indicated in 
Exercise IIb., No. i, p. 20. 

The world of Alicia / The world of Alice behind 
and Euclid j the looking-glass 




Q' * F f 

Fig. 1. 

A is the apex of a convex mirror, of large radius ; O is its 
centre, OA is the central radius or axis ; the mid-point F 
of OA is the focus. PQ is an object outside the mirror, 
perpendicular to the axis and of height a feet, P'Q' is the 
image of PQ in the mirror. Denote the various lengths as 
follows, in feet ; 


OF=FA=/; FQ'=*; Q'A=* ; AQ=y ; V'Q'=h. 
We have the following formulae : 

1 1 . 1 7 ax . 

* y f f 

The general consequences of these formulae are easy to 

Since -=-+y.we see that -> -, so that y>z. 

and that 7 <-, so that z<f or 2<AF. 

The image P'Q' is therefore always nearer the mirror than 
the object PQ is, and is never as far from the mirror as F is. 

Again, since h=Y> tne height of the image is propor- 
tional to x, its distance from F, and therefore the nearer 
P'Q' approaches F the smaller the length of P'Q', the height 
of the image, becomes. 

Life behind the Looking-Glass. 

" ' He's dreaming now,' said Tweedledee : ' and what do 
you think he's dreaming about ? ' 

" ' Nobody can guess that,' said Alice. 

" ' Why, about you ! ' Tweedledee exclaimed. ' And if he 
left off dreaming about you, where do you suppose you'd be ? ' 

" ' Where I am now, of course,' said Alice. 

" ' Not you ! ' Tweedledee retorted contemptuously. 
' You'd be nowhere. Why, you're only a sort of thing in his 
dream ! ' 

" ' If that there King was to wake,' added Tweedledum, 
' you'd go out — bang ! — just like a candle ! ' 

" ' lam real ! ' said Alice, and began to cry. 

" ' You won't make yourself a bit realer by crying,' Tweedle- 
dee remarked." 

We shall now treat Alice not as a thing in a dream, but 
as the image in a convex looking-glass of a pseudo-Alice 
who is moving about in our own world. Alice will insist 


as vehemently as she did to Tweedledee that she is a free 
agent with an independent existence, but we, looking 
from outside, will see that she conforms to the movements 
and amusements of this pseudo-Alice, whom we will call 
Alicia. We proceed to compare our (or Alicia's) observa- 
tions with Alice's own ideas about her mode of life. 

Alice's Life. 

Alicia is 4 feet tall and 1 foot broad. She starts at A 
with her back against the mirror, so that she and Alice are 
back to back, exactly the same size. Alicia is carrying a 
foot-rule which she holds against the mirror so that it 
touches and coincides with, and therefore equals, the corre- 
sponding foot-rule which Alice has. 

Alicia now walks at a steady rate of 1 foot per second 
away from the mirror, along the axis. What happens to 
Alice ? 

Suppose the radius of the mirror is 40 feet, so that 
AF=F0=/=20 feet. Then a =PQ= Alicia's height=4 ft. 

After (say) 5 seconds, AQ=v=5. 

Then---+ 7 -g+---- ? 

/. AQ'=z=4 and x=Q'F=f— z=20— 4=16. 

™™ , «* 4x16 32 
.*. P'Q'=^=7= 1 ^-=fe=3-2 feet. 

If at this moment Alicia looks round, she will notice that 
Alice has only moved 4 feet compared with her own 5 feet, 
and that Alice's height has shrunk to 3-2 feet. 

Alice's foot-rule, held vertically, has also shrunk: 'its 

1 *u • x 2. ■ 1 X * 1 X 16 

length m fact is now —f~= „- =o-8 foot. 
J 20 

Alice repudiates the idea that she has grown smaller, and 

to convince Alicia she takes her foot-rule and uses it to 

measure herself, and shows triumphantly that she is still 

exactly four foot-rules high (3-2-^0-8=4). 


Alicia also notices that Alice is not so broad as she was ; 
her breadth, in fact, has now dwindled to o-8 foot : true, 
the breadth is still equal to Alice's foot-rule, but that foot- 
rule in any position perpendicular to the axis is now only 
o-8 foot long. 
(For further numerical examples, see Exercise IIa., Nos. 1-3.) 

Contraction-Ratio Perpendicular to the Axis. 

We see that Alice continues to contract as she moves 

farther away from the mirror. The contraction-ratio in 

. h x 

any direction perpendicular to the axis is - which equals j 

and is therefore proportional to x, Alice's distance from the 
focus F. 

It is evident that Alice cannot detect this contraction, 
because Alice's ruler contracts in just the same proportion 
as Alice's body and Alice's clothes. In fact everything in 
Alice's world, regardless of what it is made, behaves in 
exactly the same way. We therefore call this contraction 
a property of space, not a property of matter. It is a 
form of influence which the space exercises on all things 
alike which enter it. And we say that one of the laws of 
space in Alice's world is an automatic contraction-ratio which 
for any direction perpendicular to the axis, is proportional to 
x, the distance from the focus. 

Contraction-Ratio along the Axis. 

Alicia now lays her foot-rule down along the axis, and of 
course Alice imitates her. 

1 1 — £ 1 W+- 

S Q A Q»S 

Fig. 2. 

Q, S are two successive points of division on Alicia's 
foot-rule, such that AQ=5 feet, QS=o-i foot, so that 



AS=5-i feet. The corresponding marks on Alice's foot- 
rule are Q', S'. We have already proved that if AQ=v=5, 
then AQ'=z=4. Further, if AS=v=5-i, then AS' =2 is 
given by 

j_I + I_ I.I _ 2Q+5-I ^25-I 
' z y f 5*1 20 5-1X20 102' 
. _, 102 

.". Ab =z=—— =4-064, approximately. 

/. Q'S'=AS'-AQ' =0-064 foot. 

.*. the contraction-ratio along the axis at Q' is 

Q'S' = o-o64 r 
QS o-i 

But the contraction-ratio perpendicular to the axis at Q' 
has been shown to be o-8. 

Since (o-8) 2 =o-64, this suggests that the contraction- 
ratio along the axis is the square of the contraction-ratio 
perpendicular to the axis, at the same place. 


Fig. 3. 

The proof of this statement is indicated in Exercise IIb., 
No. 2. Alicia therefore notices that Alice becomes thinner 


from front to back as she moves away from the mirror, 
and the rate of getting thinner is more rapid than the rate 
of getting shorter. 

For example, if Alice turns sideways and stretches 
out her left arm towards the mirror and her right arm 
away from it, the fingers of her left hand will be longer 
and fatter than those of her right hand, but the general 
effect will be to make the fingers of her right hand appear 
puffier — a chilblain effect — because these fingers have 
shortened more than they have thinned, compared with the 
other hand. 


[In this exercise, Alicia is supposed to be 4 feet high and her 
waist measurement is 1 foot broad, 6 inches thick; also /= 20 ft.] 

1. When Alicia has moved 10 feet from the mirror, show that 

Alice has only moved 6 feet 8 inches, and is now 2 feet 
8 inches tall and 8 inches broad. What happens when 
. Alice measures her height with her own foot-rule ? 

2. When Alicia has moved 20 feet from the mirror, show that 

Alice has only moved 10 feet. What is Alice's height 
and breadth in this position ? 

3. Where is Alicia, when Alice's height is reduced to 1 foot ? 

What is then Alice's breadth ? What is the length of 
her foot-rule, held vertically ? How many foot-rules 
high is Alice ? 

4. With the data of No. 1, find the thickness of Alice's waist. 

What is Alice's measurement of it ? 

5. With the data of No. 2, what is the contraction-ratio for 

Alice along the axis ? What is the connection between 
the contraction-ratios along and perpendicular to the 
axis ? 

6. Alicia is 20 feet from the mirror and holds a i-inch cube 

with its edges parallel or perpendicular to the axis. 
What is Alice holding ? 

7. When Alice's height shrinks to 1 foot, what are her waist 

measurements ? 

8. Does Alice's shape, as well as her actual size, alter as she 

moves away from the mirror ? When Alice's height has 



dwindled to i foot, a statue is made of her on an enlarged 
linear scale, 4:1. Will this statue be a good life-size 
likeness of Alicia ? 

Alice's Geometry. 

While Alicia is walking away from the mirror at a 
uniform speed, taking steps of equal length, Alice is also 
walking away in the opposite direction ; but (according to 
Alicia) Alice's steps get shorter and shorter, and so she 
advances more and more slowly ; and in fact, however far 
Alicia travels, Alice herself can never get as far as F. 
Alice of course imagines that there is no limit to the 
distance she can travel, and what Alicia calls the point F, 
Alice calls a point at infinity. If Alice walks along level 
ground, she imagines that the tip of her head and the soles 
of her feet are moving along parallel lines ; indeed the 
lines along which they move express Alice's idea of parallel- 
ism. Alicia sees that such lines actually meet at F. 

Fig. 4. 

If Alice lays a railway track along the axis, the railway 
lines will behave in the same way. 

Suppose Alicia is riding on a bicycle down the axis 
away from the mirror, how do the wheels of Alice's 
bicycle behave ? 

The contraction is greater along the axis than at right 
angles to it. Consequently, not only are the wheels of 


Alice's machine smaller than those of Alicia's, but Alice's 
front wheel is smaller than her back wheel : moreover, each 
wheel is approximately elliptical in shape, its vertical 
diameter being greater than its horizontal diameter, and 
although the wheels are turning round, the spokes which 
are vertical always appear to be longer than any of the 
others, and the spokes which are horizontal always appear 
to be shorter than any of the others ; the consequence is 
that the spokes appear to expand as they revolve from the 
horizontal to the vertical, and then to contract as they 
revolve from the vertical to the horizontal. 

Fig. 5. 

Alice herself, after careful measurement, is satisfied that 
the machine is quite normal, but Alicia will think it most 
unsteady. Space does not permit of any inquiry into the 
mechanics of Alice's life : for this, reference should be 
made to Professor Garnett's article, mentioned above. 

Innocents at Play. 

The object of this chapter is not to suggest that we are 
living in a looking-glass world, but to point out that there 
would appear to be no method of discovering the fact, if 



it were true. When Nature makes her laws of space, she 
can cast a binding spell over its inhabitants, if she cares to 
do so. But the fact that Nature is willing to answer some 
of the experimental questions which scientists put en- 
courages them to think that gradually these laws of space 
and time may be disclosed. The purpose of this chapter will 
be served if it suggests that the search is not simple, and the 
results may be surprising. 


I. BAC is a convex mirror whose radius AO is large compared 
with the object PQ. The ray of light from P parallel 

Fig. 6. 

to the axis meets the mirror at N, and is then reflected 
along the line joining N to the focus F ; the ray of light 
from P towards the centre O of the mirror is not changed 
in direction by the mirror ; therefore the image of P is 
at the intersection P' of NF and PO. Draw P'Q' 
perpendicular to OA, then P'Q' is the image of PQ. 
The focus F is at the mid-point of OA. Since the mirror 
has a large radius, its curvature is small, and NA can be 
treated as a straight line perpendicular to OA. 

FO _FP'_FQ' 

(i) Prove that 



(ii) Hence, with the notation of p. 13, prove that 

z y^f 

P'O' O'F ax 

(iii) Prove that ■jn-=|p and hence prove that h=~j. 

(iv) Hence show that x=f—z=— and that -=- F . 

V J J y y J 

If in the figure PQ moves along the axis a short distance to 
RS, and if the image of RS is R'S', and if AS=y 1 , AS>'=z 1 , 

use the formula -= I + ^ to prove that 

Q'S' ^— z _ zz ± ri _z 2 _% 2 /h\ 2 

QS — y x — y~yy~ y z P W " 

What do you deduce about the longitudinal contraction- 
ratio ? Those acquainted with the calculus should 

}iz Sv 
show that -a=-^, and interpret the result. 

What does Alicia think of the movement of the hands of 

Alicia's watch (i) when held facing the mirror (ii) when 

laid flat on the ground ? 
Alice spins a top so that its axis is vertical ; what is there 

unusual about it, according to Alicia ? 
The axis of the mirror A->0 points due east. Alice, whose 

height has become only half that of Alicia, turns and 

walks north-east. What is her direction as measured 

by Alicia ? 
Alice believes she has proved two given triangles congruent 

by the method of superposition. Does Alicia agree with 

her ? 


" The first thing to realise about the ether is its absolute 
continuity. A deep-sea fish has probably no means of appre- 
hending the existence of water ; it is too uniformly immersed 
in it : and that is our condition in regard to the ether." — Sir 
Oliver Lodge, Ether and Reality. 

The Ether. 

Those who have engaged in physical research during 
the last hundred years have been rewarded by discoveries 
of far-reaching importance and interest. Light, electricity, 
magnetism, and matter have been linked together so 
closely that it now appears that each must be interpreted 
in terms of a single medium, the ether. 
■;■ What the ether is will, no doubt, remain a subject of 
acute controversy between physicists, of rival schools for 
many years to come. Its existence was first postulated 
to serve as a vehicle for light. Experiment showed that 
light travels through space at approximately 300,000 km. 
per second, thus light takes about 8 J minutes to reach the 
Earth from the Sun. 

According to the undulatory theory, light is propagated 
in the form of a wave motion through the ether. The work 
of Weber, Faraday, Clerk-Maxwell, and others established 
the remarkable fact that electro-magnetic radiation is a 
wave motion propagated with exactly the same velocity 
as light, and therefore presumably it uses the same medium 
as a vehicle. More recent research has shown that electric 


charges are discontinuous, and that the atoms of which 
matter is composed may themselves each be resolved into 
a group of electric particles ; the group which constitutes 
each atom contains both negatively charged particles 
called electrons and positively charged particles called 
protons ; atoms differ from each other according to the 
number and grouping of these particles. Each atom may 
be regarded as a miniature ultra-microscopic solar system, 
in which the electrons describe orbits round a central 

The ether is to be regarded as a continuous medium, 
filling the whole of space, and may indeed be identified with 
space : matter, electricity, etc., are discontinuous. But 
while identifying ether with space, we must also attribute 
to it some physical qualities in order that it may serve as 
a vehicle for physical phenomena. Some physicists credit 
it with weight and density. But if Einstein's view is 
accepted, it has no mechanical properties of this nature. 
Einstein holds that the idea of motion in connection with 
the ether is meaningless; the ether is everywhere and 
always. He does not say that the ether is at rest, but that 
the property of rest or motion can no more be applied to 
the ether than the property of mass can be applied to a 
man's reflection in a mirror, although the light-rays by which 
we perceive the reflection may and indeed do possess 

Absolute Motion. 

If a passenger in a train observes another train moving 
past him, and if the motion is uniform and if there are no 
landmarks in view, it is impossible for him to determine 
whether his own or the other train, or both, are really in 
motion. This is a familiar experience. There is no 
difficulty in measuring the relative velocity of the two 
trains, but without a glimpse of the ground to act as a 


reference for measurement it is impossible to find what we 
tend to call the actual velocity of the train. 

Again, suppose two balloons are drifting past each 
other above the clouds : an observer in one balloon tends 
to think of himself at rest and the other balloon as moving 
past him. Even when he obtains an accurate ground- 
observation, he can only calculate his velocity relative to 
the earth. An astronomer might continue the work and 
tell him the velocity of the observed point of the ground 
relative to the Sun and then the velocity of the Sun relative 
to one of the " fixed stars." But even all that will not enable 
him to find his actual or absolute velocity. What reason is 
there to consider any of the stars as fixed ? we know that 
they also move relatively to each other ; what indeed can 
the word " fixed " mean at all ? Is there anything in the 
Universe we can mark down as really fixed ? Scientists 
did not like the idea that all measurements of motion must 
be relative; it seemed like building a structure of the 
mechanics of the Universe on a shifting sand. When, 
therefore, physical research demanded the existence of a 
medium filling the whole of space, the ether was welcomed 
not only for what it could do for light and electricity but 
because it appeared to offer a standard of reference for the 
measurement of absolute velocity. Scientists, therefore, 
set to work to measure the velocity of the Earth through 
the ether. The fundamental experiment which had this 
object, and which may be taken as the basis for describing 
Einstein's (restricted) theory of Relativity, was performed 
in 1887 by Michelson and Morley. We shall in future 
refer to it as the M.-M. experiment. The idea of that experi- 
ment may be easily understood by taking a simple analogy. 

Rowing on Running Water. 

A stream is flowing at 4 feet per second between straight 
parallel banks 90 feet apart. Two men start from a point 


A on one bank ; one of them T rows straight across the stream 
to the opposite bank at B and returns to A, the other L rows 
to a point C 90 feet downstream and then rows back to A. 
Each of them rows at 5 feet per second relatively to the water. 
Compare their times. 

We have called the oarsmen T and L because T rows 

transversely to the 
5 stream and L rows 




longitudinally, in the 
otjft./sec line of the stream. 
~" Now T, in order to 

reach B, must point 
his boat upstream 
along a line AP such 

A 90 ' C that if AP=5feet (i.e. 

Fig. 7. the distance he moves 

relative to the water 

in 1 second), the water will carry him down 4 feet from P 

to Q (i.e. the distance the stream runs in 1 second), where 

Q is a point on AB and ZAQP is a right angle. 

By Pythagoras, AQ 2 +4 2 =5 2 , .'. AQ 2 =25— 16=9, 
/. AQ= 3 feet. 

Therefore in each second the boat makes 3 feet headway 
along AB. 

.". the boat takes ^-=30 seconds to get from A to B. 

Similarly it takes 30 seconds to return from B to A. 

.*. the total time across and back =60 seconds. 

Now L, on his journey to C, is moving relatively to the 
water at 5 feet per second, and the water carries him for- 
wards 4 feet per second ; therefore he advances at the 
rate of 5+4=9 ^ ee ^ P er second. 

.*. the time from A to C=9-=io seconds. 



But when returning against the stream from C, his advance 
is only 5—4=1 foot per second. 

.'. the time from C to A =9°. =90 seconds. 

.". the total time downstream and up=io+ 90=100 seconds. 

/ # time of L down and up __ioo^5 

• • time of T across and back 60 3" 

It therefore takes longer to go down and up than an equal 
distance across 'and back. But the working of this example 
also shows that if we know the rate of rowing relative to 
the water, and if we know the ratio of the times taken for 
equal journeys in the two directions, we can calculate the 
velocity of the stream. 

The Michelson-Morley Experiment. 

It is known by experiment that light always travels 
through the ether at a constant rate of 300,000 km. per 
second. Let us suppose that at a 
certain moment the Earth is moving 
through the ether at a speed of u km. 
per second in the direction C->A, then 
from the point of view of a man on 
the Earth the ether is streaming past 
A in the direction A-»C at u km. per y\ q 

second. AC and AB are two rigid Fig. 8. 

equal and perpendicular rods, with 
mirrors attached at C, B, so as to face A. At the same 
moment rays of light are dispatched from A, one along 
AC and the other along AB ; these rays impinge on the 
mirrors and are reflected back to A. The motion of these 
rays corresponds to the motion of the boats in the ex- 
ample given above. Each ray travels at 300,000 km. per 
second relative to the ether-stream, since its mode of 
propagation is a wave in the ether, just as each boat 


moves at 5 feet per second relative to the water. Also 
the ether-stream is itself moving in the direction A->C 
at u km. per second, just as the water-stream is flowing 
at 4 feet per second. 

Now it takes longer to go any given distance downstream 
and back than to go the same distance across the stream 
and back. Consequently the ray from C should arrive 
back at A later than the ray from B. If then we measure 
the ratio of the times taken by the two rays, we can calculate 
the speed u km. per second of the ether-stream, and this 
is equal and opposite to the velocity of the Earth through 
the ether. 

The M.-M. experiment was designed to measure the 
ratio of these times : for a detailed account of the apparatus 
employed, reference may be made to any standard modern 
text-book on light. To the astonishment of the experi- 
menters, the race proved to be a dead heat, the ray from 
C arriving back at A simultaneously with the ray from B. 

Now the earth is describing its orbit round the Sun at 

a speed approximately of 30 km. per second, consequently 

there is a difference of speed after a 

six months' interval of about 60 km. 

per second, so that even if the Earth 

should happen to be at rest in the 

FlG ether at one moment, it could not 

be still at rest six months later. But 

the repetition of the experiment after a six months' 

interval still gave a dead heat. 

Also, in order to guard against any error arising from 
an inequality of the lengths of the arms AB and AC, the 
experiment was repeated after rotating the arms so that 
AB lay along the supposed stream and AC across it ; but 
no difference of time was detected. Further, different 
directions were tried for AB, but without any result. The 
experiment has been carried out more recently with such 


added refinements that as small a speed as ^ km. per second 
would have been detected. Here, then, was an experimental 
result which contradicted a conclusion obtained by theory. 
Clearly there was something wrong with the theory. 
Scientists were compelled to look for some explanation or 
some modification of the theory which would reconcile 
calculation with observation. 

What is the Answer to the Riddle ? 

Let us return to the illustration of the boats which 
correspond to the light-rays in the M.-M. experiment. The 
two boats start together under the conditions stated on 
p. 26, and every one is then amazed to see them arrive 
back simultaneously. How is this to be reconciled with 
the conclusions reached by calculation ? 

The first suggestion is that L rowed faster relatively to 
the water than T ; this, however, must be rejected because 
the speed of rowing through the water 
corresponds in the M.-M. experiment to li- 
the velocity of light through the ether, 
which we know is a constant, 300,000 km. 
per second. 

The next suggestion is that the courses 
are marked out incorrectly, and that 
the length of AC is less than that of AB, p IG IO> 

owing to careless measurement. But this 
view is untenable, because in the M.-M. experiment when 
the rigid arms AB'and AC were interchanged, there was 
still no difference of time. 

Fitzgerald then suggested that the arms AB and AC 
were unequal, not through faulty measurement, but 
because the shifting of a bar from a position across the 
stream to a position along the stream caused automatically 
a contraction in the length of the bar. The adventures of 
Alice have shown us that such a contraction could not be 



discovered by measurement, because the foot-rule with 
which AC is measured contracts in just the same proportion 
as the arm. 

Suppose in the example of the boats the rule which 
measures i foot across the stream contracts to § foot when 
in the line of the stream. When we measure out 90 foot- 
rule lengths to obtain AC, the outsider (Alicia) will say 
that AC is really f X 90 =54 feet, instead of 90 feet. L will 

then take ^4=6 seconds to go downstream and 54 
9 1 

=54 seconds to return upstream, so that his total time 
will be 6+54=60 seconds, which is precisely the time 
taken by T. 

This hypothetical phenomenon is called the " Fitzgerald 
Contraction." Its value depends, of course, on the velocity 
of the stream ; when the stream runs at 4 feet per second 
and the speed of the rowing is 5 feet per second, the con- 
traction ratio has been shown to be x/( 1 —%)= x /(~) 

=-|. The reader will see, from this way of writing it, 
what its value would be in other cases. 

In 1905, an alternative explanation was offered by 

Einstein's Hypothesis. 

Einstein lays down two general principles or axioms : * — 

(i) It is impossible to detect uniform motion through 
the ether. *J~>-n 

(ii) In all forms of wave motion, the velocity of propaga- I 
tion of the wave is independent of the velocity off 
th© source. ^ 

Let us consider what these axioms mean. 

(i) There is no difficulty in measuring the velocity of 
one body relative to another : all our ideas of velocity are 
essentially ideas of relative velocity, either velocities of 

- - . f\ 


other things relative to ourselves or our own velocity 
relative to something else — e.g. a man who looks at the 
road along which he is driving his car is probably estimating 
his own velocity relative to the road. But it is meaning- 
less to inquire what our velocity is relative to the ether ; 
no part of the ether can be distinguished from any other 
part : it may be possible to identify matter in the ether, 
but the ether itself defies identification. And if the ether 
cannot be (so to speak) labelled anywhere, the statement 
that a body is moving through it carries no information 
with it, or in other words has no meaning attaching 
to it. 

(ii) The second axiom is perhaps more tangible. Imagine 
an engine moving at a uniform rate along a straight railway 
line on a perfectly calm day. If the engine-driver throws a 
stone forwards, a man on the line will observe that the 
velocity of the stone is equal to the velocity given it by 
the thrower+the velocity of the engine. The faster the 
engine is moving, the faster the stone will move, although 
the thrower exerts the same effort as before. The velocity 
of the stone relative to the air, therefore, depends on the 
velocity of the source, namely, the man on the engine. 

Suppose now the engine whistles, and is heard by a man 
farther down the line. We know that the sound travels 
in the form of a wave through the air at approximately 
1100 feet per second. But the motion of the sound-wave 
is a different type of motion from that of the stone : its 
velocity of propagation through the air does not depend 
on the velocity of the engine at the moment it whistled, 
i.e. it does not depend on the velocity of the source. The 
speed of the train will affect the pitch of the sowid-wave, 
its musical note ; but the time the wave takes to reach 
the man is not affected by the rate at which the engine 
is moving. If, then, a particle in motion sends out a beam 
of light, the rate of propagation of^the light-wave through 


the ether has no connection with the velocity of the 
particle which emitted the beam of light. 

300,000 km. 

200,000 km. 

lOO.OOOkm. ! 

Fig. ii. 

P and Q represent two places 300,000 km. apart and 
rigidly connected together. I take up my position at P 
and send a ray of light along PQ and measure the time it 
takes to reach Q. If PQ is fixed in the ether (assuming 
for the moment this phrase has a meaning), the time will 
be 1 second. If, however, my observations give the time 
as (say) only § second, I can calculate that the ray itself 
only advanced §X 300,000 =200,000 km. through the ether, 
and that therefore Q must have advanced 300,000—200,000 
=100,000 km. to meet it in the same time, f second. Con- 
sequently the rigid bar PQ is moving at the rate of 
ioo,ooo4-f=i5o,ooo km. per second. But as I remain 
at P, I deduce that my velocity through the ether is also 
150,000 km. per second. This, however, contradicts 
Axiom (i), which lays down that a discovery of this nature 
is impossible. We are, therefore, forced to conclude that 
the measurement of the time of flight over this distance 
will always under all conditions be 1 second. Einstein's 
two axioms taken together, therefore, involve the following 
important result : 

Any one who measures experimentally the velocity of light 
in a vacuum will always obtain the same result (within, of 
course, the limits of error imposed by the experiment). The 
velocity of light in a vacuum, as determined by every individual, 
is an absolute constant. 


This conclusion may well cause a shock to any one who 
considers carefully what it implies ; and the shock will not 
be diminished by examining its bearing on the problem of 
the boats. 

It is, of course, important to notice the fundamental 
distinction between light-waves in the ether and sound- 
waves in the air. If an observer, when measuring the 
velocity of sound, obtains an answer which does not agree 
with the standard answer (about 1100 feet per second), he 
can at once calculate his velocity through the air. Nor is 
there any reason why he should not be able to do so ; and 
he can compare his result with that obtained by other 
methods. But with the ether it is otherwise ; the inability 
of an observer to measure his velocity through the ether 
involves the fact that his measure of the velocity of a light- 
wave must agree with the standard measurement. 

The Application of Einstein's Hypothesis. 

No one can be conscious of moving through the ether. 
An onlooker O has no difficulty in measuring the speed 
at which a man L is moving away from him ; it will be equal 
and opposite to the velocity with which L calculates that 
O is moving away from him : if each expresses his measure 
of this velocity as a fraction of the velocity of light, the 
results obtained by O and L will be numerically equal and 
opposite in sign. Relative velocities, therefore, present 
no difficulty. But O and L alike will each consider himself 
at rest in the ether and will make his own measure- 
ments on that assumption. They, therefore, must be 
regarded as looking at the world from different points 
of view. 

To explain the enigma of the boats, we must, therefore, 

consider separately the standpoint of each of the actors 

in the drama, the oarsmen T and L and an onlooker O, 

whom we will regard as poised just above T and L at the 




moment they start rowing. To make the analogy with 
the M.-M. experiment closer, imagine that the river-banks 
have disappeared, and that all we can see is an expanse 
of mter devoid of all features or landmarks — that is what 
the ether-idea requires. 

O says that this featureless ocean is 
moving in the direction A->C at 4 feet 
4 ft :j [s ec, per second ; in proof of this state- 
ment, he places a piece of cork on the 
water and observes that it at once 
(l) moves away from him in the direction 

£ AC at 4 feet per second. T and L say 
Fig. 12. that the water is motionless ; each, 

sitting in his boat, places a piece of 
cork on the water and it remains where they have placed 
it ; O, of course, says that these pieces of cork are drifting 
at the same rate as the boats. Further, T and L agree 
that O is moving away from them in the direction C->A 
at the rate of 4 feet per second ; they say that the piece 
of cork which O has dropped remains stationary, and 
that it is O who is moving away from it. 

The statement that the velocity of light is an absolute 
constant, or that each person who measures it obtains the 
same answer as any one else, when applied to the boats, 
means that T, L, and O will each obtain the same result 
when they measure the speed at which each boat is 
rowed through the water, because the boats are re- 
placing the light-rays in the M.-M. experiment. We have 
taken this common measure of the speed as 5 feet per 

Under these circumstances, our problem is to explain 
a definite experimental observation, namely, the fact that 
the boats (i.e. the light-rays) do return to A at the same 
Regard AB and AC as rigid planks of wood floating on , 


the water. T and L believe that these planks are at rest, 
just as they believe the water is at rest ; O believes that 
the planks are drifting with the water, just as T and L 
are doing. 

T, L, and O each have a foot-rule ; T holds his SBbng 
AB, and L holds his along AC. O compares his foot-rule 
with T's by actual superposition, and they note that the 
two rules agree. As long as T's rule is kept perpendicular 
to the stream, it will remain identical with O's rule ; but 
we shall see that when L, after comparing his rule with 
T's, places it along AC in the line of the stream, will 
consider it to contract although both L and T are un- 
conscious that it does so, and must remain unconscious of 
this fact, because they can have no knowledge of the 
existence of any stream carrying them along. 

With the data of the problem, T and L satisfy themselves 
by direct measurement that AB and AC are each 90 foot- 
rules long. T and L, neither of whom recognise the exist- 
ence of a stream, then calculate that their times to B, 

C respectively and back to A will be m each case — z — 

=36 seconds. And their clocks must bear this out when 
the trips have been made, for otherwise they could infer 
the influence of a stream and calculate its velocity. 

O now times T's trip. By the argument on p. 26, he 
sees that T makes a headway of 3 feet per second along 

AB and back, and therefore takes 2 x 9 ° ==60 seconds for 

the whole journey. Consequently O says that T's clock 
only registers 36 seconds when it should register 60 seconds ; 
therefore, according to O, T's clock loses. 

Now L and T take precisely equal times for their trips. 
Therefore, by O's clock, L also takes 60 seconds. But by 
the argument on p. 26, O sees that L advances at 9 feet 
per second from A to C, and returns from C to A at 1 foot 


per second. Therefore if AC =90 feet, the total time 
==^+^-==10+90=100 seconds; but the total time 
according to O is only 60 seconds. 

/. according to O, the length of AC is only _ X90 

=54 feet. 
[As a check, note that ^4+54=6+54=6o seconds.] 

It is true that L marked out AC by taking 90 of his foot- 
rule lengths ; therefore O is forced to conclude that L's 

foot-rule is only ^=f foot long, and so O says that the 

stream causes L's foot-rule, when placed along it, to contract 
to I foot. 

Further, as L also records the time of his trip as 36 seconds, 
O says that L's clock loses at just the same rate as T's clock. 
We may summarise these results as follows : 
says that (i) clocks in the world of T and L lose 
time ; they register an interval which is really 
5 minutes long as only 3 minutes (60 : 36=5 : 3) ; 
(ii) a foot-rule in the world of T and L measures 
1 foot when placed along AB at right angles to 
the stream, but only measures § foot when placed 
along AC in the line of the stream. 
T and L say that (i) their clocks keep normal time ; 

(ii) their foot-rules remain 1 foot long, in what- 
ever position they are placed. 

Who is Right ? 

It seems absurd to suggest that all of them are right. 
Let us, however, inquire what L thinks about O. Suppose 
that O and his brother O' mark out two courses, AB and 
AD, each of length 90 feet, along CA produced and AB, in 
the air just above the ocean. 


Then L says there is a current 'in the air of 4 feet per 
second which carries O and O' in the direction C-»-A ; 

4 ft. /sec. 



Fig. 13. 


O and O', of course, say that the air is at rest and that 
L is drifting with the water in the direction A->C at 4 feet 
per second. 

Suppose now that O and O' fly through the air at 5 feet 
per second (i.e. relatively to the air). O' flies to B and 
back again to A, O starts at the same time as O' and flies 
to D and back again to A. They both arrive back at A at 
the same moment. This is the experimental fact estab- 
lished in the M.-M. experiment, which needs explanation. 

It is clear that L's views (or T's views) about O and O' 
are precisely the same as those which O formed about T and 
L. The arithmetical calculations are identical and need not 
be repeated. The results may be expressed as follows : 

L and T say that (i) clocks in the world of O and O' 
lose time ; they register an interval which is 
really 5 minutes long as only 3 minutes ; 

(ii) a foot-rule in the world of O and O' measures 
1 foot when placed along AB at right angles to 
the current, but only measures f foot when placed 
along AD in the line of the current. 

and 0' say that (i) their clocks keep normal time ; 

(ii) their foot-rules remain 1 foot long, in what- 
ever position they are placed. 


It is clear, therefore, that any argument that can be 
used to support the views of or 0' can be applied with 
equal force to support the views of T and L. We must, 
therefore, regard /both views as equally true. We are 
therefore forced to conclude that each world, the world 
of O, O', and the world of T, L, has its own standard of 
time-measurement and its own standard of length-measure- 
ment. If one world is moving relatively to another world, 
their standards of time and space automatically become 

Suppose two people come together and compare their 
clocks to make sure they run at the same rate, and compare 
their foot-rules to make sure they agree, and suppose that 
afterwards they separate at a uniform rate, one from the 
other, along a line AC. Now imagine two explosions to 
take place at different times at different places somewhere 
on AC. Each observer, making proper allowance for the 
time sound takes to travel, can measure the time-interval 
between the two events and the distance-interval of the 
spots at which they occurred. But their measurements 
will not agree, either as regards time-interval or as regards 
distance-interval, for they have different standards of time 
and different standards of length. 

There is indeed one measurement about which they will 
agree, namely, the velocity of a ray of light : each of them, 
using his own clock and his own rule, will find experimentally 
that a light-wave travels at 300,000 km. per second. 

Note. — The statement (see p. 28) that further repeti- 
tions of the M.-M. experiment have confirmed the conclu- 
sion that no ether-stream can be detected requires some 
qualification. Just recently, Professor Dayton Miller be- 
lieves that he has been able to measure a drift which 
varies from zero at sea-level up to 10 km. per second at 
the summit of Mount Wilson. The details of his experi- 


ments are not yet available in this country, but Professor 
Eddington has argued {Nature, June 6, 1925) that a 
differential drift of this kind is inconsistent with astrono- 
mical observations. It, therefore, appears probable that 
Professor Miller's experiments may admit of an interpreta- 
tion different from that which he is reported to give. 


1. A man's foot-rule is really only 10 inches long ; what is the 

true length of a fence which the man measures as 12 
yards ? What will the man say is the length of a fence 
whose true length is 20 yards ? 

2. A foot-rule contracts to £ of its proper length. What is 

the true length of a line which according to this foot-rule 
is y feet ? If the foot-rule is used to measure the length 
of a line whose true length is z feet, what result is obtained? 

3. O says that two events occurred at an interval of 12 seconds 

at places 18 feet apart. What measurements are given, 
by L, if his clock only registers 45 minutes for each hour 
of O's clock, and if his foot-rule only measures 8 inches 
according to O's rule ? 

4. A stream flows at 3 feet per second, and a man can row at 

5 feet per second through the water. The width of the 
stream is 40 feet. Find the times taken to row (i) straight 
across the stream and back, (ii) 40 feet downstream and 

5. With the data of No. 4, find how far the man can row down- 

stream and back if he takes the same time as he would to 
go straight across and back. 

6. It is found that a bullet from a rifle travels 1 100 feet in the 

first second of motion. The bullet is fired along a rail- 
way line from a train at a moment when a man is 1 100 feet 
away in the line of fire. There is no wind. Does the 
bullet or the noise of the explosion reach the man first if 
the train (i) is moving towards the man, (ii) is at rest, 
(iii) is moving away from the man ? 

7. A stream flows at u feet per second, and a man can row at 

c feet per second through the water. The width of the 
stream is x feet, and the man can row straight across 


and back in the same time that he can row x x feet down- 
stream and back. Prove that — 

<T> 2 * _ *! , *i 
w *Jc 2 —u a c+u^c—W 

(ii)* 1= *</(l_" 2 ) 


" Alice looked round in great surprise. ' Why, I do believe 
we've been under this tree all the time ! Everything's just 
as it was ! ' 

" ' Of course it is,' said the Queen ; ' what would you have 
it ?' 

" ' Well, in our country,' said Alice, ' you'd generally get to 
somewhere else if you ran very fast for a long time, as we've 
been doing.' 

" ' A slow sort of country ! ' said the Queen. 'Now here, 
you see, it takes all the running you can do to keep in the same 
place. If you want to get somewhere else, you must run at 
least twice as fast as that ! ' " — Through the Looking-Glass. 

Observations at Different Places. 

If several observers, who are recording the times of 
occurrence of a series of events, wish to exchange their 
results, it is necessary for them to compare their clocks. 
Preferably the clocks should be synchronised, but it would 
be sufficient to note the difference between each clock and 
some standard clock. The standard British clock registers 
what we call " Greenwich time." 

Synchronising is a simple matter if the observers and their 
clocks are all at one place, but if the observation stations are 
far apart direct comparison is impossible, and we are forced 
to rely on indirect methods which may not be proof against 
criticism. To transport a clock from one station to another 
is not a reliable method, because the journey itself may 
set up an error in the running of the clock. The best 
method is to send signals from a standard station to all 


other stations, and use these signals to synchronise the 
various clocks or record their errors ; this in fact is done 
each day by the wireless signals sent out at noon from 
Greenwich. Wireless signals travel with the velocity of 
light, and therefore, for such comparatively small distances 
as we are concerned with on the Earth, the time of transit 
of the signal is usually negligible. But for large distances, 
such as the distance of the Sun from the Earth, the time 
taken by the signal is material, and allowance must be 
made for it in setting the clock. We shall see, however, 
that the process involves another difficulty which we are 
powerless to remove. This is best illustrated by a numerical 
example. In order to avoid big numbers and to bring the 
arithmetic of this chapter into line with that of the last, 
we shall introduce (temporarily) a new unit of length : 

60,000 km.=i leg. 
The velocity of light is therefore 5 legs per second. 

Synchronising Two Clocks. 

Suppose that two observers A and C, relatively at rest 
to each other, are at a distance of 75 legs apart, as measured 
by their own rules ; this distance is about twelve times as 
much as the distance of the Moon from the Earth. We 
shall examine the process by which A and C attempt to 
synchronise their clocks. 

5< 7- 

75 legs 


Fig. 14. 

Since light travels at 5 legs per second, A and C calculate 
that a ray of light sent by either to the other will take 

25 =15 seconds to travel across the space separating them. 


It is agreed that, at the instant when A's clock records zero 
hour, A shall send a light-signal to C, and that C, immedi- 
ately he receives it, shall reflect it back to A. 

C therefore sets his clock at 15 seconds past zero, but does 
not start it until the signal from A arrives. Immediately 
C receives the signal, he starts his clock and believes that 
it now agrees with A's clock. This opinion is shared by A, 
who, when he sees his clock indicating 15 seconds past 
zero, says to himself, "At this moment C is receiving my 
signal." From A's point of view, the fact appears to be 
established beyond doubt when the return signal from C 
reaches A at the instant his (A's) clock registers 30 seconds 
past zero. We know that A's receipt of the return reflected 
signal must occur at this instant, because otherwise A would 
be able to calculate his velocity through the ether (compare 
p. 32), and this, as we have seen, is impossible. In the 
same way, C, when he sees his clock indicating 30 seconds 
past zero, says to himself, " At this moment A is receiving 
the return signal," and this opinion is confirmed by the 
fact that, if A then reflects the signal back to C, it will reach 
C when C's clock indicates 45 seconds past zero, for the 
same reason as before. 

Now there can be no ambiguity as regards the time indi- 
cated by C's clock of an event happening to C, nor as 
regards the time indicated by A's clock of an event hap- 
pening to A. But we shall see that there is unfortunately a 
great deal of uncertainty as to the time indicated by A's 
clock of an event happening to C, or vice versa. If the 
clocks of A and C are genuinely synchronised, this uncer- 
tainty would not exist. But if there are grounds for 
suspecting that A and C are mistaken in their belief that 
they have succeeded in synchronising their clocks, there is 
no direct method of either of them ascertaining the time by 
his own clock of an event which is happening to the other. 
Although, when A sees that his clock reads 15 seconds past 


zero, he says that at this moment his signal is arriving at C, 
yet he has no direct method of making sure that this state- 
ment is true. And, by enlisting the evidence of an eye- 
witness, we shall show that there are different, but equally 
trustworthy, opinions of the time recorded by A's clock of 
the arrival of the signal at C. 

An Onlooker's Opinion. 

We now introduce an onlooker O, who considers that 
the world of A and C is moving away from him in the 
direction A->-C at 4 legs per second. 

Each individual acts on the supposition that he himself is 
at rest. In the following inquiry into O's opinions, we 
must therefore regard O as at rest and A, C as moving 
away from O. But if we had to inquire into the views of 
A or C, we should have to regard them as at rest and O 
as moving away from each of them in the opposite 

Suppose that A is passing O at the moment when A 
sends out his first light signal, and that O also sets his 
clock at zero at this instant. We can connect O with the 
world of A, C most easily by imagining that A combines 
his time-signal with a performance of the M.-M. experiment. 

A marks out a track AB at right 
angles to AC and makes it 75 legs 
long by his rule, and places a mirror 
4 legs pe r sec at B in the usual way. At the 
*"*" same time as he sends his light 

signal to C he sends another to B, 
and, as we know, both rays, re- 
0,A 75 r flected back, return to A at the 

_, same moment. 

Fig. 15. ^ T 

Now O and A agree that the 

length of AB is 75 legs, because for lengths across the 

stream their rules are identical. Also O, A, and C all 





agree that light travels at 5 legs per second through the 

Fig. 16 represents O's idea of the path pursued by the 
light signal, which is directed to the mirror at B. 

By the time the light 

signal impinges on the J^. Jj|i 

mirror at B, the arm 
AB has moved to the 
position AiB x , so that 
the signal starts from 
O, A and impinges on the 
mirror at B v and there- 
fore travels along OBj ; 
by the time it returns to 

A the arm AB has moved to the position A 2 B 2 , so that the 
return path is BjA 2 . 

The arm AB is advancing at 4 legs per second and the 
light signal travels along OB x at 5 legs per second. Suppose 
the outward journey takes t seconds. Then OB^s* legs, 
BB X =4* legs, OB =75 legs, 

/. by Pythagoras, (5*) 2 =(4*) 2 +75 2 - 
.-. 25P— 162 2 =75 2 or 9* 2 =75 2 or 32=75. 

.*. t= L ^=2S seconds. 
3 D 
.'. the total time out and back according to O's clock is 

2x25 =50 seconds. 
But by the M.-M. experiment the ray returns to A from C 
at the same moment as the ray from B. 

.*. by O's clock, the ray returns to A from C at 50 seconds 

past zero. 
But A's clock registers 30 seconds past zero when the ray 
returns to A from C. 

.*. the arrival of the ray back at A is said by O to occur 
at 50 seconds past zero by O's clock, and to occur at 
30 seconds past zero by A's clock. 


Therefore, although O's clock and A's clock agreed at 
zero hour, they do not agree afterwards: we may there- 
fore say that the synchronisation between O and A has 

Let us next ascertain O's opinion as to the time when the 
first signal reached C. 

O says that the ray from A to C is advancing at 5 legs 
per second towards a target C which is retreating at 4 legs 
per second : the ray therefore gains on the target C at 
5—4=1 leg per second. But on the return journey the 
ray advances at 5 legs per second towards a target A 
which advances to meet it at 4 legs per second : the ray 
therefore gains on the target A at the rate of 5+4=9 
legs per second. The distance which the ray has to gain 
on its target is the same on each journey (A and C 
believe this distance is 75 legs; O does not agree 
with them : but we need not stop to ascertain O's 
estimate of the distance), therefore the outward journey 
A->C takes 9 times as long as the journey back from G to 
A, so that x^ths of the total time is spent on the outward 
journey, and T l ^th of the total time on the return journey. 
Now the total time, out and back, by O's clock is 50 

.*. O says the outward journey, A->C, takes ^th of 50 
=45 seconds, and the journey back from C to A takes 
-J^th of 50 =5 seconds. 

.*. says that the ray arrives at C at 45 seconds past 
zero by O's clock. 

Also since A's clock records the total time, out and back, 
as 30 seconds, we see in the same way that says that the 
ray arrives at C at ^th of 30=27 seconds past zero by 
A's clock. 

Further, when the ray arrives at C, we know that C's 
clock registers 15 seconds past zero and is set going at this 



The occurrence of the event consisting of the arrival of 
the ray at C is therefore registered by as follows : 

O's Clock. 

A's Clock. 

C's Clock. 

45 seconds past zero. 

27 seconds past zero. 

15 seconds past zero. 

This is O's opinion of the operation. A of course does not 
agree with O ; when A's clock registers 27 seconds past 
zero, A says that it is long past the time of C's receipt of 
the signal. 

O, however, says that C's clock has been set 27—15 
=12 second-spaces behind A's clock. 

We can easily continue this process of calculating the 
times registered by O of further events. Consider the 
arrival of the ray back at A from C. 

C dispatches the ray to A and receives it back again 

2 X 7^ 
reflected from A after a total interval of — -^=30 seconds, 

by C's clock. 

Now O says that the time from C to A is only j^th of the 
total time C->A and A->C. 

Therefore O says that the ray takes ^th of 30 =3 seconds, 
by C's clock, to travel from C to A. But the time on 
C's clock when the ray left C was 15 seconds past zero ; 
therefore the time on C's clock when the ray arrives at A 
is 15+3=18 seconds past zero, according to 0. 

The occurrence of the event consisting of the arrival of 
the ray back at A is therefore registered by as follows : 

O's Clock. 

A's Clock. 

C's Clock. 

50 seconds past zero. 

30 seconds past zero. 

18 seconds past zero. 


•It is worth comparing these two events as recorded by 0. 

O's Clock. 

A's Clock. 

C's Clock. 

Event I. (arrival 

Event II. (return 

to A) 



15 seconds past 

18 seconds past 


Time -interval be- 
tween the events 



3 seconds. 

therefore says that A's clock and C's clock run at the 
same rate (each registers the interval between the two 
events as 3 seconds), but both their clocks lose time (each 
records an interval as 3 seconds which is really 5 seconds 
long) and C's clock has been set 12 seconds behind A's clock. 

What other Onlookers Think. 

Now these calculations which O has made have depended 
on the fact that the world of A, C is moving away from O at 
4 legs per second. Suppose that there is another onlooker 
P, who notes that the world of A, C is moving away from him 
in the direction A-»C at (say) 3 legs per second. Then the 
same argument which has been used to obtain O's records 
may be used to obtain P's records of the various events, 
but the arithmetic will be different, and P's opinion about 
the behaviour of the clocks of A and C will not agree 
numerically with O's opinion. P will say that A and C 
have failed to synchronise, but will form a different esti- 
mate of the amount C's clock is behind A's clock, and will 
assess at a different figure the rate at which both A's clock 
and C's clock lose. It is left to the reader to make the neces- 
sary calculations, see Exercise IV., No. 2. Each onlooker, 
therefore, has his own standard of time ; and his judgment 
of the time-interval separating two events will differ from 


that formed by another observer moving relatively to him. 
This agrees with what has been said in the previous chapter. 
But the example we have just taken shows also that it is 
impossible to synchronise two clocks which are situated 
at different places. For, although the inhabitants of the 
world in which the clocks are at rest believe that they have 
secured synchronisation, the observers in other worlds not 
only deny that they have done so, but disagree amongst 
themselves as to the amount of the difference between the 
clocks. No setting of the clocks can therefore ever secure 
general approval, or indeed approval by the inhabitants of 
more than one world. 

Simultaneous Events. 

If after A and C believe they have synchronised their 
clocks, an event takes place at A and another event takes 
place at C, and if each records the time of the event which 
has happened to himself, and if these two records are the 
same, then A and C will say that the two events happened 
simultaneously. But with the data of our example, we 
see that O will say that the event at A took place before 
the event at C, for according to O when A's clock reads 
27 seconds past zero C's clock reads 15 seconds past zero. 
Therefore if A and C both say that the times of the events 
are 27 seconds past zero, O says that, when the event occurs 
at A, C's clock has only got as far as 15 seconds past zero 
and therefore the event at C, timed as 27 seconds past 
zero at C, has not yet occurred. In fact the time-interval 
between these two events is 27— 15 =12 seconds as measured 
by the clock-rate of A or C, which is equivalent to 20 seconds 
as measured by the clock-rate of O, for 5 of O's seconds 
are the same as 3 of A's seconds or C's seconds. O will 
therefore say that the event at A took place 20 seconds 
(by O's clock) before the event at C. 

A and C therefore call two events simultaneous which 


O considers occur at a definite time-interval, and other 
onlookers will agree with O in saying that the events are 
not simultaneous, but will disagree with O as to the length 
of the time-interval between them. It is therefore im- 
possible to attach any meaning to a general statement 
that two events at different places occurred at the same 
time. If the time-standard of one world makes them 
simultaneous, the time-standard of other worlds require 
a time-interval between the events. Since there is no 
reason to prefer the opinion of one onlooker to that of any 
other, we cannot say that any one opinion is more correct 
than any other. The bare statement that two events at 
different places were simultaneous is, therefore, devoid 
of meaning, unless we also specify the world in which this 
time-measure has been made. 

Union of Space and Time. 

Time by itself ceases to be an absolute idea ; it is a pro- 
perty of the world in which it is measured, and each world 
has its own standard. 

Each individual has, of course, his own time-rule and 
his own distance-rule which he thinks of as absolute, 
because he thinks of his own world as at rest. But in a 
sense this is a delusion, because a transference to another 
world will modify each of them ; a change in time-measure 
is bound up with a change in distance-measure. As we 
have already seen, the onlookers do not agree with A, C, 
or each other as to the distance between the two places 
where the events occurred, any more than they agree with 
the time-interval between the events. To quote the 
celebrated phrase of Minkowski : " From now onwards 
space and time sink to the position of mere shadows, and 
only a sort of union of both can claim an independent or 
absolute existence " — i.e. an existence to which all onlookers 
will give equal recognition and apply equal standards of 


measurement. We shall see later what form this union 


1. A and C measure their distance apart as 50 legs ; an on- 

looker P notes that the world of A, C is moving away from 
him in the direction A->C at 3 legs per second. A passes 
P at zero hour by A's clock and P's clock, at which 
moment A sends a light signal to C in order to synchron- 
ise with C ; this signal is reflected back to A. What is 
P's estimate of the times recorded on the three clocks of 
(i) the arrival of the signal at C, and (ii) the return of the 
signal to A ? 

2. Repeat No. 1, if A and C are 75 legs apart. 

3. With the data of No. 1, D is a person in the world of A and 

C at a distance of 100 legs from A and on the other side 
of C from A. If A and D synchronise, find the difference 
between their clocks according to P in terms of second- 
spaces (i) on A's clock, (ii) on P's clock. 

4. Repeat No. 1, assuming that the world of A, C is moving 

away from P in the direction C->A at 3 legs per second, 
P and A being as before at the same place at zero hour. 

5. With the data of No. 1, if an Event I. occurs at A and an 

Event II. occurs at C, and if A and C describe these 
events as simultaneous as recorded by their own clocks, 
which event will P consider to have occurred first ? 
Repeat this problem with the data of No. 4. 

6. Two events, I., II., occur simultaneously at different places 

in the world of A and C. An onlooker O says that I. 
occurred before II. Would it ever be possible for some 
other onlooker to say that II. occurred before I. ? 

7. With the data of No. 1, find P's estimate of the distance of 

A from C. 



"The progress of Science consists in observing inter- 
connections and in showing with a patient ingenuity that the 
events of this ever-shifting world are but examples of a few 
general relations, called laws. To see what is general in what 
is particular, and what is permanent in what is transitory, 
is the aim of scientific thought." — A. N. Whitehead, An 
• Introduction to Mathematics. 


We have shown in the previous chapters, by means of 
numerical examples, that any eye-witness will consider 
that standards of measurement of distance and time vary 
from one world to another. The real nature of these 
variations cannot be appreciated unless we pass on from 
numerical illustrations to general formulae. We therefore 
shall now proceed to express in algebraic form the relations 
betweei^ two worlds which are moving with uniform 
velocity relatively one to another. These formulae may 
then be utilised for solving special numerical cases. 

It will simplify the work if we introduce a new unit of 
length : 

300,000 km. (i.e. 5 legs) =1 lux. 

The velocity of light is therefore 1 lux per second. 

Statement of the Problem. 

It may assist the reader if we state in great detail the 

problem proposed for solution in this chapter. 



The world of A and C is moving away from O in the 
direction A-^-C at a uniform velocity of u luxes per second ; 
at the instant when A passes O, both A and O set their 
clocks at zero hour. A and C are at rest relatively to each 
other, and they measure their distance apart as x x luxes. 
A and C believe they have synchronised their clocks. 

An event (Event I.) occurs at A at zero hour by A's 
clock ; another event (Event II.) occurs at C at t x seconds 
past zero by C's clock. Therefore in the world of A and C 
the distance-interval between the two events is x x luxes, 
and the time-interval between the two events is t x seconds. 
There is complete agreement between A and C as to both 
of these interval measurements. Each regards both him- 
self and the other as at rest in the ether. Their distance- 
measures agree because they can use the same rule to measure 
out AC ; their time-measures agree, because otherwise 
they could deduce the velocity of their common world 
through the ether. 

Next consider O's point of view. He says that Event I. 
occurs at O at zero hour, and that Event II. occurs at C 
at (say) t seconds past zero by his own (O's) clock. O regards 
himself as at rest and A, C as moving away from him. 
O therefore takes the distance-interval between the two 
events as the distance of C from him at the moment when 
Event II. takes place. Suppose that O's measure of this 
distance is x luxes. Then says that the distance-interval 
between the two events is x luxes and the time-interval 
between the two events is t seconds. 

In short, the interval between the two events is registered 
by A or C as x x luxes, t x seconds, and by O as x luxes, 
/ seconds. 

What are the formulae which connect x, t with x x , t x ? 

Before tackling this general problem, we shall ascertain 
O's opinion about the measuring-rule used by A or C, the 
running of their clocks, and their attempts to synchronise. 




A marks out a length AC of x x luxes along the line of 
motion of A's world relatively to O. What is the length 
of AC, according to O ? 

B B, B 2 

v luxes per sec. 


Fig. 17. 


Suppose that O is watching A performing the M.-M. experi- 
ment. A and C agree that the lengths of AC and AB are 
each x x luxes, O agrees that the length of AB is x % luxes, 
but says that the length of AC is different, say z luxes. 

says that the arm AB moves away from him at u luxes 
per second, so that the ray sent towards the mirror at B 
impinges on it when AB has moved into the position 
AjBi ; the path of the ray is therefore AB r Similarly, 
the ray returns to A when AB has moved into the 
position A 2 B 2 , so that the return path is B^. 

Suppose the time from A to B x or from B x to A 2 , 
is k seconds by O's clock. O makes the following 
calculations : 

ABi=^ luxes (light travels along AB X at 1 lux per second). 
BB ± =ku luxes (AB advances at u luxes per second). 
AB =x x luxes (O agrees with A's measurements across the 
.*. by Pythagoras, k 2 =k?u 2 -\-x 1 2 
:. W—Wu*=x? or k 2 (i-u 2 )=x 1 z 


I— M a 


The total time by O's clock from A to B and back is 
2k seconds. 

.*. the total time by O's clock from A to C and back is 
2k seconds. 
But O can also reason as follows : 

From A to C, the ray travels at i lux per second towards 
a target C, z luxes away, which is retreating at u luxes per 
second. Therefore the ray gains on the target at (i— u) 
luxes per second. 

.*. the time from A to C by O's clock is seconds. 

i — u 

Similarly from C to A the ray travels at I lux per second 

towards a target A, z luxes away, which is advancing at 

u luxes per second. Therefore the ray gains on the target 

at (i+w) luxes per second. 

.*. the time from C to A by O's clock is — ; — seconds 


.'. the total time from A to C and back is — — + — ^— seconds 

I — U I-j-M 
z(i+u)+z (i— U) _Z+ZU+Z— ZU 2Z , 

- (i- w )(i +w ) f=^2 =£Z^r* seconds. 

But the total time by O's clock is 2k seconds. 


.*. i _ m2 =2& or z=k(i— u 7 ) 
;. 2 2 =& 2 (i-w 2 ) 2 , but k 2 =^~ 2 

••• * a =^2(l-« 2 ) 2 =*! 2 (l-« 2 ) 

.". z=x 1 V(i— u 2 ). 

Therefore says that a length in the direction of motion 
which A and C measure as x x luxes is really x x V{l— u 2 ) 
luxes : 

Or, in proportion, what A and C measure as I lux is in O's 
opinion really V(i— u 2 ) luxes. 


Now V(i— u 2 } must be less than i ; consequently says 
that the measuring-rule used by A and C, when placed 
along the line of motion, contracts ; and the contraction- 
ratio is V{i— m 2 ). 

Comparison by O of A's Clock and C's Clock with O's 

Our numerical examples have shown that every one 
will agree that A's clock runs at the same rate as C's clock. 
The reason for this may be stated as follows : 

An essential feature in every argument is that each 
individual regards himself as at rest in the ether and that 
all the observations he makes must bear this out. He 
cannot make any measurement which will reveal his 
velocity through the ether. A and C agree that their 
distance apart is x x luxes : they therefore argue that a 
time signal sent from either to the other and reflected 
back will return after 2X X seconds, and their clocks must 
bear this out. But the experiment in which A sends a 
signal to C and receives it back again is identical with the 
experiment in which C sends a signal to A and receives it 
back again. Both A's clock and C's clock record the time 
of this experiment as performed by each of them as 
2% seconds. Therefore A's clock and C's clock must run 
at the same rate. We have seen from numerical examples 
that O admits this, but says that both clocks lose and that 
they have not been synchronised. Let us now calculate 
O's estimate of the time-difference between A's clock and 
C's clock. 

In order to synchronise the clocks, A proposes at zero 
hour by his clock to send a light-ray to C. As they agree 
that AC is x x luxes, they calculate that the signal will take 
x x seconds to reach C. Consequently C sets his clock at 
x x seconds past zero and starts it at the instant the light-ray 


Now we have just seen that by O's clock the time from 

A to C is - — m seconds, and the total time out and back is 
i — u 


.^—^2 seconds. O therefore says that the fraction of the 
total time out and back occupied by the journey out is 

Z _^_ 2Z _ Z (l-\-u)(l—u) _ L-\-U 

I — U ' I—U 2 ~~I — U 2.Z 2 " 

Now A's clock registers 2x x seconds for the total time out 

and back. Therefore O says that, when the ray arrives 

i I ii 
at C, A's clock registers — — X2x 1 =x 1 (i-\-u) seconds past 

zero. But at this instant C's clock starts off at x ± seconds 
past zero. 

.*. A's clock is ahead of C's clock by x x {~l-\-u)—x x seconds 
=x 1 -\-x 1 u— x 1 =x 1 u seconds. 

Therefore when A and C think they have synchronised 
their clocks, O says that A's clock is x x u second-spaces 
ahead of C's clock. 

The difference between the clocks depends on the value 
of x v the length of AC. Therefore the farther C is away 
from A in the direction of motion of the world of A, C 
from O, the more A's clock is ahead of C's clock, according 
to O. Suppose, for example, the world of A, C is moving 
due east away from O. Then A's clock is ahead of any 
clock east of A and is behind any clock west of A. Both 
these results are expressed in the statement given above, 
because, if C is west of A, x x is negative, and a clock which 
is a negative number of seconds ahead of another clock is, 
of course, a positive number of seconds behind it. 

We must therefore regard each place on the line of 
motion of AC as having its own clock : the inhabitants 
of the world of A, C think all these clocks are synchronised, 
but O says each registers a local time whose difference 


from that of A is given by the formula above. We may 
express the facts by a diagram showing the local time, 
according to O, of the instant when A is passing O, 

C x 2 luxes 

u luxes per sec. 
— » 

x, luxes 

x 2 u sec. after zero A x,u sec. before zero 

zero hour 


Fig. 19. 

which is taken as zero hour both by A and O. The 
distances indicated in the diagram represent A's or C's 


A and C record the time-interval between two events 
as 1 second. What is O's estimate of this time-interval 
by his own clock ? 

With our previous notation, we know that O says that 
the time from A to C and back is 2k seconds by O's clock, 

where & 2 == x 2 or k= ,. x 4|2V 
1— u 2 v(i— u z ) 

But A says that the time from A to C and back is 

2% seconds by A's clock and O must agree with him. 

.*. O says that 2x x seconds on A's clock measures the same 
time-interval as 2k seconds = //t *<,< seconds on O's 

V (I— ur) 

.*. O says that (in proportion) 1 second on A's clock measures 

the same time-interval as —tj- o\ seconds on O's 

v(i— u*) 


It is important to remember that this is a statement of 

O's view about the behaviour of A's clock. 


Since V(i— u 2 ) is less than i, / / 2 ^ is greater than i , 

and therefore says that A's clock loses. But, of course, 
A equally says that O's clock loses. Our results always 
depend on the point of view from which the progress of 
events is being observed. 

Time and Distance Intervals between Two Events. 

The data which determine the two events have been 
stated in great detail on p. 54. The diagram represents 
O's view of the events. 

(I.) Position at zero hour — 


-» it luxes per sec. 

(II.) Position at t sec. past zero by O's clock — 

1 . 


Fig. 20. 

Event I. occurs at zero hour at A at the instant when A is 
passing O. Event II. occurs at C at t seconds past zero by 
O's clock. A and C say that the distance-interval between 
the two events is x x luxes, i.e. their measure of AC is x x luxes. 
O says that Event I. occurred at O, and that Event II. 
occurred at the position of C at t seconds past zero by O's 
clock. O therefore says that the distance-interval between 
the two events is x luxes, which is his measure of the length 
of OC in (II.). O also says that the measure of OA in (II.) is 
ut luxes, because A is moving away from him at u luxes per 

.*. O says that by his rule AC=x—ut luxes. 

Now A measures AC as x x luxes and O says that a measure- 
ment of 1 lux by A is really -v/(i— w 2 ) luxes (see 


p, 56). Therefore O says that AC is really x x V(i— u 2 ) 


.'• x 1 V(i—u 2 )=x—ut 


This relation is very important. 

Again, suppose that the time of Event II. at C is recorded 
by C's clock as t x seconds past zero. Then A and C must 
agree that the time-interval between the events is t x seconds. 

Now O says that A's clock is x x u seconds ahead of C's 
clock (see p. 58). Therefore O says that, when Event II. 
occurs, the time on A's clock is t x +x x u seconds past zero. 
But we know that 1 second on A's clock measures the same 

time-interval as ^,*_ u *\ seconds on O's clock. 

.*. when Event II. occurs, the time on O's clock is 

J^t^L seconds past zero ; but the time on O's 
V(i— u 2 ) r 

clock is t seconds past zero, 
. , t x +x x u 

" *-V(l-M 2 ) 

x — ut 
;. t x +x x u=tV(i-u 2 ) ; now %= ^ (l _^ 2) 

. t W/ T „2\ UJX-Ut) t(l-U 2 )-u{x-Ut) 

" ^l=^(I-w 2 )- v(I _ w2) V(I-1*) — 

_t— uH— ux-\-u 2 t 
~ V(i-u 2 ) 
_ t—ux 

''• tx ~V{l.-U 2 )- 

This relation is also of great importance. It may be a 
help to the reader if we re-state what has been established. 
Two events occur at distance-interval x x luxes and time- 
interval t x seconds according to A, C, and at distance-interval 


x luxes and time-interval t seconds according to 0. The 
world of A, C is moving away from the world of O at u luxes 
per second, and distances are measured as positive in the 
direction of motion of A from O. Then A's records are 
connected with O's records by the formulae : 

_ x—ut _ t—ux 

If then we know the distance interval and the time-interval 
between two events as recorded in one world, we can cal- 
culate the distance and time-intervals between these two 
events as recorded in any other world moving with uniform 
velocity relative to the former, along the line joining the 
two events. 

A's Opinion of O's Records. 

It has been pointed out frequently in previous chapters 
that there is no observer whose records are entitled to more 
respect than those of any other observer. It is therefore 
essential to show that the formulae just obtained are con- 
sistent with this view. Using the same notation and axes 
as before, A says that O is moving away from him at (— u) 
luxes per second. Now A says that the distance and time- 
intervals between the events are x x luxes and t x seconds. 

.*. the formulae just obtained show that 

O's distance-interval should=- J 7 7 ^ ^=-77 V\ 

v(i— w 2 ) V(i— u 2 ) 

and O's time-interval should =^-7/ 4r= A — ^ 

V(i— w 2 ) a/(i— m 2 ) 

.*. the formulae just obtained should be equivalent to 

_ x x -\-ut x t _ t x +ux x 

X -V{I-U*) anCl r ~A/(l-M 2 )- 

Unless they' are, there is not that reciprocal relation 


between O and A which the Theory of Relativity requires. 
We may state the problem as follows : 

x—ut , t—ux 

Given thatx 1 = ^^ 1 __ uV j and h=^( 1 ^tf) 

Phw that x^ v x ^._ uK) and *= V(l _^. 

a;— ut u(t—ux) 
(i) We have *i+«<i= v < (i __ < ^ +V(i- < ^ 

#— ut+ut— u 2 x _ x(i—ti?) 
- V(i-u 2 ) ~V(i-u 2 ) 


•'• *-</(l-M 2 )' 

(ii) We have <i+«* 1 = V( I _ < ,2) + ^(i-^ 2 ) 

f— «3g+«x— -«* a ^ __ l(i — u 2 ) 
= V(i-w 2 ) V(i - w 2 ) 

=tV{i-u 2 ) 
t x -\-ux x 

V(l-M 2 )* 

2 = 

We therefore see that the relations which express A's 
opinion of O's world are consistent with, and can be deduced 
from, the relations which express O's opinion of A's world. 

The Velocity of Light. 

The formulae which connect the two worlds introduce 
the expression V(i— w 2 ), which is imaginary if «>i, i.e. if 
the velocity of one world relatively to the other is greater 
than the velocity of light. We therefore say that we can 
have no experience of a body moving with a velocity greater 
than that of light. And in all our results u must stand for 


a fraction between +i and — i. It is customary to repre- 
sent V(i-u 2 ) by i or to put P= v ,*__ uS , so that #>i. 
In this case, the standard formulae may be written : 

% 1 =P(x—ut) ; t 1 =ft(t—ux) 

x=&{x x +ut^ ; t=/3{t 1 +ux 1 ) 

where P=-tt- «>*• 

V(i—u 2 ) 

And the results on pp. 59-60 may be stated as follows : 
(i) O says that the length of a line in the direction of 
motion which A measures as 1 lux is ^ luxes. 

(ii) O says that a time-interval which A's clock records 
as 1 second is /3 seconds. 


1. The world of A is moving at § lux per second due east from 

O. What is O's opinion about (i) the length of A's 
foot-rule, (ii) the rate of running of A's clock ? What is 
A's opinion about O's foot-rule and O's clock ? 

2. A and C, who are relatively at rest at a distance apart of 

5 luxes, have synchronised their clocks ; the world of 
A, C is moving away from O in the direction A->C at 
ft lux per second. A passes O at zero hour by O's 
clock and A's clock. What does O say is the difference 
between A's clock and C's clock ? D is a place in the 
world of A, C, such that DA = 10 luxes, DC = 15 luxes. 
What does O say is the difference between D's clock and 
A's clock ? What does O say is the time recorded by the 
clocks of A, C, D when O's clock records 25 seconds past 
zero ? 

3. With the data of No. 1, A records two events as happening 

at an interval of 5 seconds and at a distance apart of 
3 luxes, the second event being due east of the first 
event. What are the time and distance-intervals of the 
events as recorded by O ? 


4. With the data of No. 3, solve the question if the second 

event is due west of the first event. 

X— Ut . Xx + Utx 

5. Given that *i=- ,/ — ; and* = . , 

t—ux , t 1 +ux 1 
Prove that U= . — =^andf = <7 v 

1 VI M a VI— M a 

6. If Event I. is the dispatch of a light-signal by A and 

Event II. is the receipt of the light-signal by C, show 
that with the usual notation (i) x 1 =t 1 , (ii) x=t. What 
does this mean in terms of O's opinion ? 

7. Using the equations on p. 63, prove that x*—t* is always 

equal to x x 2 - tj 2 . 


" ' That's the effect of living backwards,' the Queen said 
kindly ; ' it always makes one a little giddy at first, but there's 
one great advantage in it, that one's memory works both 

"'I'm sure mine only works one way,' Alice remarked. 
' I can't remember things before they happen.' 

" ' It's a poor sort of memory that only works backwards,' 
replied the Queen." — Through the Looking-Glass. 

The position of an event in history is fixed if we know 
(i) when, and (ii) where it took place. These necessary data 
must be expressed with reference, at any rate implicitly, 
to some standard event. The " when " is usually referred 
to the Christian Era, t years a.d. ; the " where," if 
on the Earth's surface, may be described in terms of 
longitude and latitude, the standard being the Greenwich 
meridian and the- Equator. Astronomers often state 
the " where " of an event in space in terms of Right 
Ascension and Declination, together with some distance 

Any event is, therefore, fixed by recording its time- 
interval and its space-interval from some actual or hypo- 
thetical standard event. Now we have seen that both 
these records vary with the world in which they are observed. 
To say that the battles of Waterloo and Hastings occurred 
at an interval of 749 years is an intelligible statement if 

addressed to people on this Earth. It would convey no 



meaning, or in fact a false one, to some one in a world moving 
rapidly along the line joining Hastings and Waterloo. If 
these two events had been recorded in this other world, 
the time-interval (and the space-interval) would be quite 
different. It is obviously desirable to try to discover 
some property connecting the two events whose measure 
will have the same numerical value, in whatever world 
records of the events are made. If this can be done, we 
can regard such a property as something absolute, inde- 
pendent of all observers, retaining the same value however 
it is viewed. Measures of time-intervals and distance- 
intervals are not absolute in this sense : they vary from 
world to world. But |nere is a kind of union of the two 
about which opinions do not change, a union which will 
be measured with equal magnitudes by all observers. We 
shall first consider a numerical example to show what the 
nature of this union is. 

Records of the Interval between Two Events by Various 

A's world observes two special events and notes that 
Event II. occurs 12 seconds later than Event I. and at a 
distance 4 luxes due east of it. The points at which 
Events I., II. occur will be called E, F respectively. 

A writes his record of the interval between the events 
in the form (4 ; 12). With our previous notation, this is 
short for %=4 ; t x =i2. 

(i) O is an observer who says that A's world is moving 
away from him at f lux per second due east. How does 
O record the interval between the events ? 

We have the relations 

x x +ut x t x -\-ux x 

x -V(z—u?)' l ~V(i-u*) 

where (x ; t) is O's record and x x =/\, ^=12, w=§. 


V(i-« 2 ) = v^(i-A) = ^(M)=I 
4+f X 12 _2o+36 _56 

f 4 4 

and , = Hdl2<_4 = 6o±H = e =l8 
t 4 4 

.*. O records the interval between the events as 14 luxes 
and 18 seconds, or, more shortly, (14 ; 18). 

(ii) P is an observer who says that A's world is moving 
away from him at f lux per second due west. How does 
P record the interval between the events ? 

Using the same formulae as before, we now have 

%=4> ^i= I 2, «=— f 
... V(i-^) = v/(i-^ T ) = v / (if)=| 

. , 4+(-f)*g _ gQ-36 ^ = __ A 

' ' — 4 A A 

and , = H±tiL4 = 6o=i2 = l8 =I2 
t 4 4 

/. P records the interval as —4 luxes and 12 seconds, or, 
more shortly, (—4 ; 12). 
This means that P says Event II. occurs 4 luxes west 
of Event I. 

(iii) Q is an observer who says that A's world is moving 
away from him at ± lux per second due east. What are 
Q's records ? 

Here %=4, t x =ia, w=| 

.-. v-(i- w 2 HV(i-M)=v-(A)=f 

. 4+(?)i2 ^ 2Q+48 _68 
. . x — 3 3 3 

i 3 3 

.*. Q records the interval as 

68, 76 .__„ /68. 76 s 


68, 76 , ft 

— luxes, '— seconds, or ( - 
3 3 v 



We will set down the records of three other observers, 
leaving the calculations to the reader, 
(iv) R is an observer for whom u=— f. 

Show that R's record is ( ; — Y 

V 3 3/ 
(v) S is an observer for whom w=i§. 

Show that S's record is (■— ; -~\ 

(vi) T is an observer for whom w=—- 2 r_. 

Show that T's record is ( - ; — ). 
\3 3/ 

Let us now collect these results in a single table, arranging 

them so that the distance-intervals are in ascending order 

of magnitude, apart from sign. 










Value of u luxes per second 


_ S 



_ 4 







Distance-interval, x luxes . 
Time-interval, t seconds . 










This table shows that if in passing from one world to 
another the distance-interval (apart from sign) is increased, 
then the time-interval is also increased. 

What relation connects the values of t and x ? A 
glance at the table does not suggest any obvious relation. 
But if we write down corresponding values of t 2 and x 2 , 
it is not difficult to guess the answer. This is done below : 

Thus for R we have t=\^%, x=— 9 J ; 

therefore 2 2 =(i4f) 2 =^X^=^=2i5i, 
and^={- 9 i) 2 =|xf=^=8 7 i. 






6 4 i£ 


t 2 
* 2 







f 2 -* 2 








In each case, we see that t 2 — % 2 =i28. 

Although the time-interval t seconds and the distance- 
interval x luxes varies from one world to another, all 
observers alike agree that the value of t 2 — x 2 is 128. 

We shall represent the expression t 2 — x 2 by s 2 , so that 
in the special case above we have s 2 =i28 or s=V(i28) 
=11-3, as measured by every observer : and we say that 
n-3 measures the separation of the two events. This 
name is due to Professor Whitehead. 

The separation of two events defined by the formula 

S 2=*2_#2 

is an entirely new conception ; it is neither time nor distance, 
but some kind of fusion of the two. Its importance arises 
from the fact that it is independent of the world in which 
the records are made ; all observers attribute to it the same 
numerical measure, provided it is agreed to measure time 
in seconds and distance in luxes. The separation, therefore, 
represents something absolute, some intrinsic property 
connecting the two events, without regard to the conditions 
under which the events are observed. 

Formal Treatment of Separation. 

If, with our previous notation, A records the interval 
between two events as x 1 luxes, t x seconds, and if O records 
the interval between the same events as x luxes, t seconds, 
then we can prove that t 2 — x 2 =t x 2 — % 2 . 


Using the formula x= *} +ui * • t= *}+ UXl 
we have t*-x*J^±^-&l±^ 

l—U 2 l—U 2 

_ (t 1 2 +2ut 1 x 1 -\-u 2 x 1 2 ) - {xf+zut^+uH*) 
I— u 2 

_ t 1 2 +2Ut 1 X 1 +U 2 X 1 2 —X 1 2 —2Ut 1 X 1 —U% 2 

l—U 2 
= ^(I_ W 2)_ % 2( I _ W 2) 

l^U 2 

=t 2 -x 2 . 
If then we represent t 2 —x 2 by s 2 , we see that tf—xf 
also equals s 2 , and therefore the value of s as calculated 
by one observer is the same as the value of s calculated 
by any other observer. Consequently the separation 
between two events, as defined above, survives transforma- 
tion from one world to another. An expression of this 
kind is called an invariant. Its measure has nothing to 
do with the circumstances of the observer; it represents 
an objective relation between, or a physical property of, 
the events themselves. 

Real and Imaginary Values of the Separation. 

In the numerical example discussed above (p. 68), the 
value of t 2 — x 2 was positive, and therefore the value of s, 
its square root, was real. But if t is less than x, then 
s 2 =/2_^2 ==a negative^number ; in this case s is the square 
root of a negative number (as for example V~ Y6) and 
is therefore imaginary. This does not of course mean 
the events are imaginary, but merely that the measure of 
the property, called the separation of the events, is ex- 
pressed in some cases by an imaginary number. It is 
easy to see when this will happen. 

Suppose Event I. occurs at E and Event II. occurs at 
F, and that the interval is x luxes, t seconds. Then a ray 


of light will take x seconds to move from E to F. Suppose 
now the time-interval between Events I. and II. is less 
than the time a light-ray takes to travel from E to F, then 
t is less than * and therefore s 2 =t 2 — x 2 =a negative number. 
Therefore the separation is imaginary if the time-interval 
between the events is less than the time required to send a 
light signal from one place to the other : in other words, if 
the separation is imaginary it is impossible for a message, 
sent off from the place E where Event I. has occurred, 
describing that event, to arrive at the place F where Event II . 
occurs before the actual occurrence of Event II. This 
fact will be recognised by all observers alike, for if one 
observer obtains an imaginary valufe for the separation, 
so will they all. 

If, however, s is real, t is greater than x. Consequently, 
if as soon as Event I. has occurred at E a wireless signal 
reporting the event is dispatched to F, every one will agree 
that it will reach F before Event II. occurs. 

The intermediate case when s is zero is easy to interpret. 
This requires that t=x or that the time-interval between 
the events is equal to the time a light-signal takes to travel 
from the one place to the other. Suppose that Event I. 
is the dispatch of a light-signal from E, and that Event II. 
is the receipt of this light-signal at F, then t=x and the 
separation is zero. The Sun is 93,000,000 miles, or 500 luxes, 
away from the Earth. If Event I. occurs on the Sun 
and if Event II. occurs after a time-interval of 8 minutes 
20 seconds (=500 seconds) on the Earth, then the separa- 
tion between these two events is zero. 

It is important to remember that when an observer puts 
on record the time of an event, he does not give the time 
at which he sees the event take place, but the corrected 
time after allowing for the time the light-ray has taken 
to reach him — i.e. he records the actual time at which he 
believes the event took place, not the time at which he 


saw it happen. For example, the observed time of an 
eclipse of one of Jupiter's moons would depend on how 
far away Jupiter was from the Earth when the eclipse was 
observed. To find the time at which the eclipse really 
happened, allowance must be made for the time the light- 
ray took to reach the Earth. It was the discrepancy 
between the calculated times and the observed times of 
the eclipses of Jupiter's moons which first enabled a Danish 
astronomer, Roemer, in 1675, to calculate the velocity of 

Time-Order of Two Events. 

Suppose that O records the interval between Event I. 
at E and Event II. at F as x luxes, t seconds, where x and 
t are each positive, so that Event II. occurs after Event I. 
and distances in the direction E to F are measured as 
positive. Suppose another observer A is moving relatively 
to O in the direction E to F at u luxes per second, so that 
m is also positive. A records the time-interval as t x seconds. 
Then we know that 

* 1_ V(i-w 2 )- 

If t x is positive, A says that Event II. occurs after Event I. ; 
but if t x is negative, A says that Event II. occurs before 
Event I. 

Is it conceivable that, after all proper corrections (as 
noted above) have been made, O and A should disagree 
as to the order in which the events took place ? The 
answer is bound up with the nature of the separation 
between the events. 

(i) An Imaginary Separation. 

If the separation is imaginary, t 2 is less than x 2 , and 
therefore, as each is positive, t is less than x. 

Put t=vx so that v<i. 



' t—ux vx—ux _ x(v—u) 
We thenjiave *i= ^^-y^.^-^.^- 
Now u can never be greater than 1 ; no observer can 
move faster than a light ray. But u can have any value 
between o and I, because we can think of A as moving 
away from O with any velocity up to 1 lux per second. 
.*. since v<i, we can think of an observer A who is 
moving away from O with a velocity greater than 
v luxes per second, i.e. so that u>v. 

But if u>v, v— u is negative 
.*. t x is negative 
.'. this observer A says that Event II. occurs before Event I. 

Again, since v<i, we can choose an observer B who is 
moving away from O with a velocity exactly equal to v luxes 
per second, i.e. so that u—v. 

But if u—v, v—u=o 

/. ^=0 , 
.\ this observer B says that Event II. occurs simul- 
taneously with Event I. 

Again, we can obviously choose an observer C who is 
moving away from O with a velocity less than v luxes per 
second, i.e. so that u<v ; then v—u is positive 

.*. t x is positive 

/. this observer C agrees with O that Event II. occurs 
after Event I. 

Now all these observers are equally entitled to their own 
views. Their time-records are of course actual as opposed to 
observed times — that is to say, the observer takes the time 
of his clock when he sees the event happen, and corrects 
for the time the light-ray has taken to reach him. In this 
way he obtains what he calls the actual time at which the 
event took place. But as no preference can be given to 
any one observer over any other, we are forced to conclude 


that it is meaningless to attach a time-order to events whose 
separation is imaginary. For such events we may say, 
using comparative terms, that whereas in slow-moving 
worlds Event II. occurs after Event I., in fast-moving worlds 
Event II. occurs before Event I., and there is one world 
in which the Events are actually simultaneous. Another 
descriptive form of this statement is based on the idea 
that the occurrence of any event cannot be said to be 
caused by any event which follows it. If some observers 
say that Event I. occurs before Event II., while other 
observers with equal justice say that Event II. occurs before 
Event I., then it is impossible to imagine that there is any 
causal relation between the events. Now this is what 
happens when t is less than x, i.e. when the time-interval 
between the events is less than the time a light-ray takes to 
travel from one place to the other. If, then, we substitute 
for " causal relation " the word " force," and if we say that 
neither event can exert a force on the other event, we are 
really saying that no force can be propagated with a 
velocity greater than that of light. If, for example, the 
Sun is regarded as exerting a gravitational force on the 
Earth, the propagation of this force takes at least 500 
seconds to reach the Earth from the Sun. We shall, 
however, see later that Einstein's theory abolishes the idea 
of a gravitational force altogether. 

(ii) A Real Separation. 

If the separation is real, t 2 is greater than x 2 , and there- 
fore t is greater than x. 
Put t=vx so that v>i. 

Then, as before, t-,=~r, 4r. 

1 v(i— u 2 ) 

Now u can never be greater than 1, but v is greater than 1 ; 
.". v>u 
.'. t x is always positive. 


Every observer will therefore agree with O that Event II. 
occurs after Event I. 

Consequently, if the separation is real, the time-order of 
the two events is definite, the same for every one. 

We know that different observers make different time- 
sections of the Universe : but all observers alike will place 
Event I. in a time-section which precedes their time-section 
containing Event II. For each of them, Event I. appears 
as a feature in the story leading up to Event II., and may 
therefore be said to contribute, however indirectly, to the 
occurrence of Event II. We may say that when the 
separation is real there is some causal connection between 
the events. 

Proper Time. 

Suppose with our previous notation A records two events 
which happen to him. He says the time-interval is t x 
seconds, and the disfence-interval is zero, for both events 
occur at the same place, namely, the place where he is. 
The separation between the events is given by s i =t 1 2 — o 2 
=t x 2 or s=<j. 

Now consider an observer O who says that A's world is 
moving away from him at u luxes per second : suppose 
as usual that Event I. occurs at zero hour when A is passing 
O. Then O records the interval between the events as 
x luxes, t seconds. The following diagram exhibits the 
two views : 

A' s opinion 

0' s opinion 

Event I 
Event II 












Fig. 21. 



Since both events happen to A, says that the distance- 
interval x luxes equals ut luxes ; x—ut. 

O also says the separation is given by s 2 =t 2 — x 2 

:. t 1 2 =s 2 =t 2 —x 2 =t 2 —uH 2 =t 2 (i—u 2 ) 

.: t x =w{i—u 2 ). 

Now t x seconds is the time-interval recorded by A, the 
person to whom the events happen. It is the time accord- 
ing to which each individual records the incidents of his 
own life, and it is called the " Proper Time " for the indi- 
vidual concerned. Since V(i— u 2 ) is less than 1, we see 
that t x is less than t, and consequently the proper time 
between two events is less than the time-interval recorded 
by any other observer. If we consider the life-history of 
an individual, the measure of the time-interval between two 
events will depend on the observer ; but if we regard the 
individual as carrying his own clock about with him, we 
call his measure of the time-interval the " proper time," and 
it is less than that of any one else. He is in fact measuring 
the separation between two events wholly in time, while 
other observers measure it. partly in time and partly in 
distance. An observer whose measure takes more distance 
must also automatically take more time as well, in com- 
pensation. The separation between two events is equal to 
the proper time between them, that is, the time-interval 
as measured by the person to whom the events happen. 
Of course if the separation is imaginary it is impossible for 
the two events to happen to the same person, and " proper 
time " ceases to exist. 

Exercise vi 

1. With the data on pp. 68, 70, verify that the records of 

R, S, T are as given in the text. 

2. With the data on p. 68, can you find an observer who will 

say that the events occurred at the same place ? If so, 


what velocity does this observer attribute to A, and 
what is his measure of the time-interval ? 

3. A gives the interval between two events in the form #1=3, 

*i = 5- O says that for A's world u=^g. What are O's 
records for the interval ? What is the separation ? 

4. A records the interval between two events as 5 luxes, 

13 seconds; O records the time-interval as 15 seconds. 
What is O's record of the distance-interval ? What 
is the separation ? What velocity does O attribute 
to A ? 

5 . O says that Event I. at E is given by * = 2, x = 5 , and Event II. 

at F is given hy t' —6, x' = 12 ; units being seconds and 
luxes. What are the time- and distance-intervals as 
recorded by O ? If A is moving away from O in the 
direction E to F at $ lux per second, what is the time- 
interval according to A ? Interpret your result. What 
is the separation ? 

6. O says that two events happen at different places at the 

same time. What can you say about their separation 
and their time-order ? 

7. O says that two events happen at different times at the 

same place. What can you say about their separation 
and their time-order ? 

8. With the data of No. 5, can you find an observer for whom 

the events will be simultaneous ? 

9. On the same day the following events are noted : 

Event I. Earthquake at Tokio at 12 noon. 

Event II. Formation of a Sun-spot at 12*06 p.m. 

Event III. Disappearance of the Sun-spot at 12.12 p.m. 
What can you say about the time-order of these events ? 
[O. I get out of bed at 7 a.m. and retire to bed at 10 p.m., 
Greenwich time ; an observer says that my bed has 
moved 72,000 luxes in the interval. How long does he 
say I have been out of bed ? If I have moved about 
with the velocity of light all the time I have been out of 
bed, how long do I say the time is ? 



" There is no difference between Time and any of the three 
dimensions of space except that our consciousness moves along 
it. . . . The civilised man can go up against gravitation in a 
balloon ; why should he not hope that ultimately he may be 
able to stop or accelerate his drift along the Time dimension, 
or even turn about and travel the other way ? " — H. G. Wells, 
The Time Machine. 

Hitherto we have only considered events which occur at 
points in a straight line along which the worlds are separ- 
ating. It is easy to extend what has been said so as to 
include events occurring at any points in space. 

Points in a Plane. 

If we take two perpendicular lines in the plane, we can 
fix the position of any point in the plane by stating its 
distances from these 
two lines. For ex- 
ample, take a point O 
and draw from it a 
line Ox due east and 
a line Oy due north. 
Suppose any point A 
is 5 miles east and 3 miles north of O ; this fixes the position 
of A. If we start at O and walk 5 miles #-wards, i.e. 
eastwards to M, and then walk 3 miles jy-wards, i.e. north- 
awards along MA, we arrive at A. The distances 5 and 3 
are called the co-ordinates of A ; we say that the point A 
is given by #=5, y=% or that its co-ordinates are (5, 3). 


O 5 m x 

Fig. 22. 


The #-co-ordinate is always put first. Suppose that B is 

another point in the plane and that its co-ordinates are 

(9, 6), then we can move from A to B by going a farther 

distance 9—5=4 miles awards and 6—3=3 miles y-wards. 

In the figure, AK=4, KB =3 

/. AB 2 =AK 2 +BK 2 =4 2 +3 2 . 

In the same way, if the distance-interval between A and B 

is X units awards and Y units ^-wards, so that AK=X and 

BK=Y, we have 

AB 2 =X 2 +Y 2 . 

The co-ordinates of a point are simply its distance- 
intervals from O measured (i) awards, (ii) ^y-wards. If we 
know the co-ordinates of any two points A and B, we can 
find the distance-intervals of B from A measured awards 
and ^y-wards by subtracting the #-co-ordinates and the 
jy-co-ordinates, as above. 

Suppose now an observer O says that the world of A, B 
is moving away from him in the direction Ox at u luxes per 
second, and that an Event (I.) occurs at A and another 
Event (II.) occurs at B after an interval T seconds. Then 
O says that the interval between the events is T seconds, 
X luxes awards, Y luxes y-wards. Now in the world of A, 
the time-interval is different, say T x seconds ; and the 
^-interval is different, say X 2 luxes ; but the ^/-interval, 
say Y x luxes, is the same, because the rules of O and A 
agree when across the stream, therefore Y=Y r 
Now we have already proved that T 2 — X 2 =T X 2 — Xj 2 

/. since Y=Y 1 , T 2 -X 2 -Y 2 =T 1 2 -X 1 2 -Y 1 2 . 
But if O measures the length of AB as r luxes and A 
measures the length of AB as r x luxes, we know that 
X 2 +Y 2 =r 2 and X 1 2 +Y 1 2 =f 1 2 . 
.-. T 2 -f 2 =T 1 2 -r 1 2 . 
We therefore put s 2 =T 2 -X 2 -Y 2 =T 2 -f 2 , and call s 
the separation between the events whose intervals are 



(X, Y ; T) according to O and (X lf Y x ; T^ according to A, 
and whose space-intervals are r, r x according to O, A. 

Points in Space./ 

To fix the position of any point in space, we take 
three mutually perpendicular planes, called the planes of 
reference, and measure the distance of the point from 
each plane. 

Suppose, for example, we draw on a horizontal plane a 
line Ox due east, a line Ov due south, and a line Oz vertically 
upwards. A point A in the air could be fixed by saying it 
was 4 miles east of O, 3 miles south of 0, and at a height of 
2 miles above O. If we start at O and walk 4 miles awards, 
i.e. eastwards to P, and then walk 3 miles .y-wards, i.e. south- 
wards along PM to M, and then rise 2 miles 2-wards, i.e. 
vertically upwards along MA, we arrive at A . The distances 
3, 4, 2 are called the co-ordinates of A ; we say that the 
point A is given by x =4, y=3, z—2, or that its co-ordinates 
are (4, 3, 2). The co-ordinates are always written in this 
order, x, y, z. 

Fig. 23. 

Now suppose we move from A to any point B by travel- 
ling X units #-wards along AK to K, and then Y units 


^-wards along KH to H, and then Z units 2-wards along 
HB to B. We say that the distance-intervals of B from A 
are X, Y, Z. 

Now ZAKH is a right angle, 

/. AH 2 =AK 2 +KH 2 =X 2 +Y 2 . 

Also ZAHB is a right angle, 

.'. AB 2 =AH 2 +HB 2 =X 2 +Y 2 +Z 2 . 

This shows how the length of AB can be calculated as 
soon as we know the distance-intervals of B from A, and as 
before we can calculate these, by subtraction, if we know 
the co-ordinates of A and B. 

Suppose now an observer O says that the world of A, B 
is moving away from him in the direction Ox at u luxes per 
second, and that an event (I.) occurs at A and another event 
(II.) occurs at B after an interval T seconds. 

Then O says that the interval between the events is 
T seconds, X luxes %-wards, Y luxes ^-wards, Z luxes 
2-wards. Suppose A says that these intervals are respec- 
tively T x seconds, X lf Y x , Z x luxes. Then we know that 
although T and X are not equal to T x and X lt yet 
T 2 — X 2 =T! 2 — X x 2 . 

Further, Y=Yj and Z=Z X , because the rules of A and O 
agree when put across the stream. 

/. T 2 — X 2 — Y 2 — Z 2 =T! 2 — Xi 2 — Yi 2 — Zj 2 . 

Suppose O and A measure the length of AB as r luxes and 
r x luxes respectively. 

Then r»=X 2 +Y 2 +Z 2 and r 1 2 =X 1 2 +Y 1 2 +Z 1 2 
/. T 2 — r 2 =T 1 2 — rf. 

We therefore put s 2 =T 2 -X 2 -Y 2 -Z 2 =T 2 -r 2 , and call 
s the separation between the events whose intervals are 
(X, Y, Z ; T) according to O and (X^ Yj, Z x ; Tj) according 
to A, and whose space-intervals are r, r x according to O, A. 

We have now applied the idea of separation to any two 


events occurring at any points of space and time, and have 
thereby constructed a function or expression which may 
properly be called a physical reality. 

Four-Dimensional Space-Time. 

In order to specify the time and space-intervals between 
two events, we see that it is necessary to give values to 
four different variables, each of which can change inde- 
pendently of the others, namely X, Y, Z, T. 

Although each observer distinguishes sharply between 
space and time, the distinction drawn by one observer is 
not the same as that drawn by another. What one 
measures as " time," another measures partly in space 
and partly in time. The distinction, therefore, between 
space and time is subjective — that is to say, the observer, 
although unconsciously, is affected by his circumstances in 
his discrimination between the two. We cannot therefore 
suppose that this distinction corresponds to an objective 
physical reality. And so we are forced to the conclusion 
that we live in a four-dimensional world, divided arbitrarily 
by each observer into three dimensions of space and one 
dimension of time, but in reality an entity to be called 

Owing to the fact that people on the Earth never move 
relatively to each other with really high velocities, the time 
and distance axes of any one person agree fairly well with 
those of any other. Life would be a perplexing and em- 
barrassing matter if this agreement did not exist. But 
the distinction between space and time drawn by an in- 
habitant of a Beta particle moving with a velocity approach- 
ing that of light is so widely different from that of the 
physicist observing him that any kind of common social 
life would be unthinkable. 

Every observer maps out the Universe with his own 
space and time axes. Suppose an observer A were to 


classify a large number of events all of which occurred at 
the same moment. Then this collection of events forms a 
time-section or time-cleavage by A of the Universe. Now 
events which are simultaneous for A are usually not simul- 
taneous for O. Suppose O takes one event in A's list and 
proceeds to make a catalogue of events simultaneous with 
it. Then we know that O's list will not agree with A's list : 
in other words, O's time-cleavage of the Universe is not the 
same as A's ; they may have some members in common, 
but speaking in general terms there will be far more dis- 
agreement than agreement. 

The Universe is to be regarded as a collection of events, 
anywhere and anywhen, an entity which mathematicians 
call a continuum, and the difference between O and A is 
simply that they slice it up differently. The Universe as 
an entity is time-less (and space-less). What each indi- 
vidual perceives is merely his own time-section. History 
records some of the time-sections of our ancestors, and 
Mr. H. G. Wells forecasts time-sections of our descendants. 
With neither group have we the power to obtain direct 
acquaintance, merely because we cannot put ourselves 
into the position in which the desired time-cleavage would 
be the natural one. But all events, past, present, and 
future as we call them, are present in our four-dimensional 
space-time continuum, a universe without past or present, 
as static as a pile of films which can be formed into a reel 
for the cinematograph. It is obviously absurd to attempt 
to form a picture of a four-dimensional Universe, but it 
may suggest ideas if we consider the structure of a three- 
dimensional Universe whose inhabitants partition it by 
two axes of space and one axis of time. 


Suppose a worm can crawl anywhere on a vast horizontal 
plane surface, but has no power of raising itself out of the 




Fig. 24. 

plane or burrowing into it ; and, better still, imagine that 
the worm has no idea that there is even such a thing as 
" above " or " below." In that case the worm is living in 
a three-dimensional con- 
tinuum, two dimensions 
in space and one dimen- 
sion in time, forming his 
space-time world. 

Draw two perpen- 
dicular straight lines ox, 
oy on the plane, and draw 
a. line ot perpendicular to 
the plane to represent the 
time-axis. The worm, of 
course, cannot picture this 
line ot any more than we 

can visualise a fourth axis of reference in addition to three 
mutually perpendicular axes chosen for our space-axes. 

Suppose the worm starts its life at the point a, co- 
ordinates x x , y x , at a time an observer O at rest at records 
as t t seconds past zero. Then O denotes this event by the 
point A in space whose co-ordinates are x v y lt t v As the 
worm moves about on the plane, each event in its life is 
represented by a point in space. Finally, the worm dies 
at b, co-ordinates x 2 , y 2 , at a time O records as t z seconds 
past zero. Then O represents this event, the worm's 
death, by the point B in space, co-ordinates x i} y 2 , h- The 
whole history of the worm from birth to death is then 
represented by a curve in space starting at A and ending 
at B. And we say, using Minkowski's phrase, that the 
curve AB is the " world-line " of the worm. Suppose now 
that there are numerous worms and other objects in the 
plane. Each has its own world-line, and if we take the 
complete collection of all these world-lines, they constitute 
the space-time worm Universe. The collision of any two 


objects, in fact the happening of any event, is recorded by * 
an intersection of two world-lines. If O compiles a cata-, 
logue of simultaneous events, he is simply making a time- 
section of these world-lines, and is choosing points on the 
world-lines which are at the same height above the plane 

Worms who move slowly will agree with O's conclusions. 
But a rapidly moving worm R will make such different 
time and space measurements that the world-lines he 
constructs will (so to speak) be quite a different shape from 
O's. The shape, however, hardly matters. If two of O's 
world-lines intersect, the corresponding world-lines of E 
must also intersect, because any intersection represents a 
space-time event. Consequently, if we think of O's world- 
lines as forming a vast network in space, then R's world- 
lines will also form a network. The meshes in the one 
system may be quite different in size and shape from those 
in the other ; but to each mesh and each corner of that 
mesh of O's network there will correspond a unique mesh 
and a unique corner of that mesh of R's network. 

Straight World-Lines. 

World-lines can clearly be of any shape ; given the 
observer, the shape is an index of the worm's history. 
What meaning must be attributed to the statement that a 
world-line is straight ? 

Choose the origin O so that the worm is born at O at 
zero hour and dies at P, co-ordinates X lf Y x at T 2 seconds 
past zero according to the observer O. Draw PC parallel 
to the time-axis and of length T x units ; draw PM per- 
pendicular to Ox so that OM=X x , MP=Y r 

Then O and C represent in space-time the birth and 
death of the worm. Suppose the worm's world-line is 
the straight line OC. Let K, any point on OC, repre- 
sent an event in the life of the worm, and suppose the 



co-ordinates of K are X, Y, T; in our figure, ON=X, 
NQ=Y, QK=T. 

Fig. 25. 

From similar triangles, rp = OP == pM : 

Our co-ordinates measure the distance and time-intervals 
since the birth of the worm. 

: OM 

' * T 1 "~OP~Y 1 Xj- 

This means that the time-intervals and the space-intervals, 
measured from the birth of the worm, all increase at the 
same uniform rate ; and it also follows that the separation 
of an event from O increases at the same uniform rate. 

The observer, therefore, says that a worm, whose world- 
line is straight, is moving with uniform speed in a straight 
line : on the Newtonian Theory, the worm would be 
moving freely, uninfluenced by any force. 

A numerical example may help to make these ideas 
more intelligible. 



Let us suppose that the progress of the worm is repre- 
sented by the following table, the units of time and space 
being seconds and luxes : 
















^-co-ordinate X 










y-co-ordinate Y 










In this table the space- and time-intervals measured 
from the start all increase at the same uniform rate. 

Take, for example, the interval between the events 
D and E ; 

T=52— 39 =I 3 ; X=i2— 9=3 ; Y=i6— 12=4 
.*. the space-interval of E from D= V(X 2 +Y 2 ) = ^(9+16) 

the separation of E fromD = V(T 2 — X 2 - Y 2 ) = -/(I69— 25) 
= v / i44= I2 . 

We should obtain the same numerical results for any 
other pair of successive events. 

Further, the space-interval of K from A is the sum of 
the space-intervals A to B, B to C, C to D, . . . H to K ; 
and the separation of K from A is the sum of the separa- 
tions B from A, C from B, D from C, . . . K from H. 
For the space-interval of each portion is 5, and there- 
fore the sum of the eight space-intervals is 8x5=40 ; 
while the space of interval of K from A is 

v / 24 2 + 322=^576 + 1024= ^1600=40. ' 
Similarly, the separation of each portion is 12, and there- 
fore the sum of the separation for the eight intervals is 
12x8=96 ; while the separation of K from A is 

Vio4 a — 24 2 — 32 s = ^10816— 1600 = ^9216 =96. 


Curved World-Lines. 

The fact that if a line is straight, the distance of the 
end-point from the starting-point is equal to the sum of 
the lengths of the various portions of the line is a funda- 
mental fact in ordinary geometry. We have now seen that, 
correspondingly, if a world-line is straight the separation 
of the last event from the first event is equal to the sum 
of the separations between successive events measured 
along the line. 

If a line is curved , the distance of the end-point from the 
starting-point is less than the sum of the lengths of the 
various portions of the curved line. In contrast to this, 
we shall show that if a world-line is curved , the separation 
of the last event from the first event is greater than the 
sum of the separations between successive events measured 
along the world-line. 

A world-line is associated with the history of an actual 
particle ; it represents the movement of the particle as 
viewed by some observer. The speed of the particle 
cannot, therefore, exceed the velocity of light. If the 
interval between any two events in the history of the 
particle is measured by T seconds, R luxes, we know that 
T> R and therefore the separation is real. In the numerical 
example, given above, if T=i3 we have R=5- 

Now consider a worm whose 
world-line consists of two straight 
portions AB and BC. Suppose the 
events A, B, C are recorded by O as 
follows : 

A, t=o, x=o, v=o ; 

B, 1=13, x=2, v=5 ; 

C, *= 2 6, *=6, v=8 ; FlG - 26 - 

the units being seconds and luxes. 


Then the intervals are 

A to B, ^=13, X x =2, Y lS =5 

B to C, T a =26-i3=i3, X 2 =6-2=4, Y 2 =8-5=3 

••T 1 " I 'X i -T-*»Y 1 -»" 
.*. the co-ordinates do not increase at equal rates, and so 
the world-line ABC is not straight. 
The separation of B from A is 

■v/( I3 2_ 2 2_ 5 2) = v / (l69 _ 4 _ 25 ) = V(I40) . 

The separation of C from B is 

a/(i3 2 -4 2 -3 2 ) = ^(169-16-9) = ^(144) =12. 
The separation of C from A is 

</(26 2 -6 2 -8 2 ) = ^(676-36-64) = V (576) =24. 
But \/(i4o) is less than 12 
.*. the separation of B from A+the separation of C from 
B is less than the separation of C from A. 
We see, then, in this case that if the world-line ABC is 
not straight, tfee separation of C from A is greater than 
what we may call the separation measured along the world- 
line ABC. 
We shall now give a general proof of this result. 

Straight and Curved World-Lines. 

If the world-line ABC of a particle is straight, we know 
that it moves with uniform velocity in a straight line. 
This involves two things : 

Suppose the time and space-intervals from A to B are 
T x seconds, r x luxes, and the corresponding intervals from 
B to C are T 2 seconds, r 2 luxes. 

Then (i) the space-interval of C from A is 7i+f 2 luxes. 

(ii) ^r =q^, for these fractions represent the speed 
of the body from A to B and from B to C. 


It is, of course, true that the time-interval of C from A 
is T x +T 2 seconds, but this is true whether the world-line 
is straight or curved, if we are dealing with the progress 

of a body. 

Now if either of the conditions (i) and (ii) ceases to be 
true, the world-line is no longer straight ; the portion AB 
will no longer be in the same straight line with BC. 

Condition (i) holds only if the space-movement of the body 
is a straight line ; condition (ii) holds only if the body 
moves at a constant speed. Each condition is necessary 
for uniform velocity. Thus if the body describes a straight 
line at variable speed, (i) holds and (ii) fails, and if the 
body describes (say) two sides of a triangle in the plane 
with constant speed, (i) fails and (ii) holds. In each case 
the world-line is not straight. Of course if the body 
moves in a plane curve with variable speed, both con- 
ditions fail. 

Separation measured along a World-Liije. 

If the world-line AB of a body is straight, with the 
above notation, we know that the separation of B from 
A is VCIV*— R x 2 ). We can represent 
this geometrically by drawing a right- 
angled triangle abH, so that ab=T 1 , 
aH=Rj, and AaRb=go°. 

Then by Pythagoras, &H 2 +R 1 2 =T 1 2 , 
or fiH^T^-Rx 2 , 

.-. 6H=V(T 1 2 -R 1 2 ). 
The side 6H therefore measures the separation. 

Now suppose the world-line AC of a body is curved. Take 
a large number of successive events in the life of the body, 
D, E, F, G, . . . N. If we take a sufficient number, each 
portion AD, DE, EF, . . . of the world-line is nearly 
straight, and we can then find the separation, as above, 



for each of these portions. If we add together all the 
separations for the small portions, we say that the sum 
is the separation of C from A measured along the world-line. 
We shall prove that this is less 
than the separation of C from A. 
To do this, it will be sufficient to 
show that if AB and BC are two 
different straight world-lines, then 
the separation of C from B+the 
separation of B from A is less 
than the separation of C from 
A. For we can then argue as 
follows : 

The separation CA is less than the 
sum of the separations CD, DA ; 
which is less than the sum of the separations CE, ED, DA ; 
which is less than the sum of the separations CF, FE, ED* 

and so on. 

Fig. 28. 

Maximum Separation. 

Suppose the world-line of a body consists of two different 
straight portions AB, BC, and suppose the time and space- 
intervals of B from A are T x seconds, r x luxes, and that 
the corresponding intervals of C from B are T 2 seconds, 
r 2 luxes. Suppose also that the corresponding intervals 
of C from A are T seconds, r luxes. Then T=T!+T 2 ; 
and r cannot be greater than r x +r 2 . Since AB is not in 

a straight line with BC, we know that either ^ is unequal 

to j or that r x +r 2 is greater than r. Possibly both these 

conditions hold, but at least one must be true. 

We shall represent the separations of B from A and 
C from B by the method shown above. 



Draw triangles aHb, 6KC with &K parallel to «H, and 
so that aR=r lt ab=T v Z.aUb =90° 

and bK=r 2 , bc=T 2 , £bKc=go°. 
Produce cK and «H to cut at N, then 6KNH is a rectangle. 

The separations of C from B and B from A are represented 

by cK and 6H, but 6H=KN 
.*. the separation of C from B-f the separation of B from A 

Now the separation of C from A= \/[T 2 — r 2 ] 
= ^[(T 1 +T 2 ) 2 -r>] 

Now if t^ is not equal to ~r, abH. and bcK are triangles of 

different shapes, so that the angles ball, cbK are unequal : 
consequently ab is not in line with be 

.*. ab-\-bc is greater than ac or T 1 +T 2 >oc. 
Also r cannot be greater than r x +r t and r 1 +r a =aH+&K 
.*. the separation of C from A =\ / [{T 1 +T i ) 2 —r 2 ] 
>A/[ac 2 -(f 1 +y 8 ) 2 ] or V[ac*-aW] 
>cN since ac 2 =aN 2 +Nc 2 
> separation of C from B+separation of B 
from A. 


But if ^ should equal ~^, so that ab is in line with be, 

then yj+^2 must be greater than r. 

Now if ab is in line with be, ac=ab-\-bc=T 1 +T 2 

.*. the separation of C from A 

= ^[(Ti+Ta) 2 -^] = V[ac*-r*] 

> A /[flc 2 -(/' 1 +f 2 ) 2 ] or V[ac*-aW\ 

>cN as before 

>separation of C from B+separation of B from A. 
Consequently, if a world-line from A to C is not straight, 
the separation of C from A measured along the world-line 
is less than the direct separation of C from A — that is to 
say, is less than the separation measured along the straight 
world-line joining A and C. 

The Geodesic Law of Motion. 

Newton's first Law of Motion states that the progress 
of a body under the action of no forces is represented by a 
straight world-line. We can now state this law in more 
general terms. 

If a body is moving freely, and if A and C are two events 
in its history, the space-time path followed by the body between 
A and C is such that the separation of C from A measured 
along that path is a maximum. 

The striking feature of this statement is thatrftj- specifies 
a path which is unique, and is independent of the axes of 
reference or what is the same thing of the observers. There 
are an unlimited number of paths joining two points in 
space-time, but there is one of these which stands apart 
from all the rest in virtue of the fact that all observers 
alike agree that it yields a separation greater than the 
separation obtained by measurement along any other 
path. The world-line which has this unique property is 
called a geodesic. 

It is interesting to note that if the body were to follow 


a curved line from A to C travelling at the same speed as 
a light-ray, the separation along each portion of the curve 
would be zero, since T=r for a light-ray, and therefore 
the total separation measured along this curved world-line 
would also be zero. We can, in fact, unite C with A by 
a curved world-line along which the separation can have 
any assigned value from the maximum down to zero, but 
the geodesic path is the only possible path for a body 
moving freely. 

The only objective property we have so far discovered 
is this property called " separation." Time and space by 
themselves merely express relations between the observer 
and the thing observed : separation is the physical reality. 
Now in the previous chapter we saw that the separation 
between two events in the life of a body is equal to the 
"proper time " for that body — that is, the time-interval 
measured by a clock which the body carries about. We 
may therefore express the conclusions of this chapter by 
saying that bodies in the Universe, if left to themselves, 
follow the path which makes the proper time between their 
birth and death as great as possible. A body chooses the 
path which gives (in its own view) greatest length of life. 
If A and B represent any two events in the life of the 
body, the path from A to B is so chosen that its passage 
occupies tike maximum amount of time (according to its 
own clock). 

This rule of conduct has been called by Mr.. Bertrand 
Russell the " Law of Cosmic Laziness " ; it is the substitute 
Relativity has made for the inclination of a body to follow 
" the line of least resistance." 


1. An event A is given by # — 1, y=3, £=8, 2=35, and an 
event B is given by x =4, y = 7, z = 20, t = 1 20, units being 
luxes and seconds. What is the separation of B from A ? 



2. With the data of No. i, the world -line of a particle is the 

straight line AB. What are the space co-ordinates of an 
event happening to the particle when ^=52? What are 
the co-ordinates of the event happening to the particle 
for £ = 14 ? 

3. With the data of No. 1, another observer says that the 

events A and B both occur at the same place. What 
time-interval does he attribute to the two events ? 
What is the nature of a straight world -line joining A 
and B, as judged by this observer, and how is his time- 
reckoning described ? 

4. What is the general appearance of the world -lines of 

(i) Trafalgar Square Tube Station, (ii) a train on the 
Bakerloo Tube, as observed by (a) a porter on the 
platform, (6) the engine-driver, (c) a person in the Sun. 

5. The position of a particle in space- time is given by #=27**, 

y = 36a, z=6oa, t = &$a, and the history of the particle is 
obtained by giving a successively all values from o to 4. 
What can you say about its world-line ? What is the 
separation between the events corresponding to a =4 and 
a=o ? 

6. Three events A, B, C in the life of a particle which moves 

in a plane are given by t=o, x=o, y=o ; t = i?, #=6, 
y = i5; 2 = 34, at = 18, y =24. What is (i) the separation 
of C from A, (ii) the sum of the separations of C from B 
and B from A. Is the particle moving freely ? 


"We recognise certain varying states or conditions of matter, 
and give one state one name and another another as though it 
were a man or a dog. Of matter in its ultimate essence and 
apart from motion we know nothing whatever. As far as we 
are concerned, there is no such thing." — Samuel Butler, 

Composition of Velocity. 

Suppose that in the world of A, B a body C starts from A 
at zero hour (both by A's clock and O's clock) and moves 
at a uniform rate to B, and 
according to A travels the 
distance AB=# 1 luxes in t x 
seconds. Suppose A measures 
the velocity of C as v luxes per 



Then «=3. °>A B 

** Fig. 30. 

Suppose also that O says 
that the world of A, B is moving in the direction A->B at 
u luxes per second. What velocity does O assign to C ? 

O says that the journey to B occupies t seconds and is of 
length x luxes where (see p. 63) 

*i+^i , . t 1 -\-ux 1 
x= -7 =r and /= , 4,- 

Vl-M 2 VI— u 2 


:. says the velocity of C is j luxes per second where 

x_ x 1 -\-ut 1 _ t x ' __ v-\-u 
t~ U-\-ux^~~ , UX-i~T-\-UV 


We may state this result as follows : 

If a body C is moving at v luxes per second in the world 
of A along AB, and if the world of A is moving away from O 
in the direction A->B at u luxes per second, the velocity of 

C as measured by O is — r — luxes per second. 

This result disagrees with the Newtonian method of 
composition of velocities. If a man is in a train travelling 
at 20 feet per second, and if he throws a bottle forward in 
the direction of motion of the train at 10 feet per second, we 
should expect a workman on the line to say that the bottle 
started moving at 20+10=30 feet per second. The for- 
mula just proved shows that it is not strictly accurate 
to combine the velocities by simple addition. But of 
course, for such small velocities as these, the correction is 

inappreciable . Approximately 1 foot per second = — 9 luxes 

per second 

2 1 

.*. the two velocities are — § and --g luxes per second 

— +— 
io 8 io 8 

.'.the composite velocity is luxes per second. 

1 2 
I+ io" 

10 8 

X io 9 feet per second. 

I- r-r 16 

IO 1 


feet per second. 

which is indistinguishable from 30 feet per second. 


But for particles moving at very high speeds, such as, 
for example, fi particles emitted by radioactive substances, 
which can move with velocities as high as 0-99 luxes per 
second, the correction introduced by this formula is 

It is interesting to take the extreme case where C is moving 
with the velocity of light, i.e. v—t. 

The Newtonian law of composition would make O assess 
the velocity of C at i+m luxes per second. But the Rela- 
tivity formula gives -x~=i lux per second, so that O's 

measure is identical with A's measure. This result is of 
course only a repetition of the statement which lies at the 
root of the whole theory — all observers alike who measure 
the velocity of light must obtain the same result. 

Transverse Velocity. 

Suppose, next, the body C starts from A at zero hour, and 
moves at a uniform rate at right angles to AB at w luxes per 
second. What velocity does O assign to C ? 

Suppose A says that C travels y t luxes in t x seconds. 

Then w="~. 
O says that C travels y luxes in t seconds, where 

y=y t and *=v=^- 
But x 1 =o, since A says that C moves at right angles to 



O says that C moves with a velocity composed of 
m>Vi_ m 2 luxes per second at right angles to AB, and 
u luxes per second along AB. 


Lastly, suppose that A assigns to C a velocity of v luxes 
per second along AB, and w luxes per second at right angles 
to AB. After t x seconds by A's clock, C has moved x x luxes 
in the direction AB and y x luxes in the direction perpendic- 

ular to AB, where v=~ and w=^. 
t i h 

O says that t seconds have elapsed, and that C has moved 
x luxes and y luxes in these two directions, where 

x,-\-uL t-A-ux x 


:. O says that C's velocity along AB is j luxes per 
second, where as before 

x_ x 1 +ut 1 __ v-\-u 
t~t 1 -{-ux 1 ~i-\-uv 


and that C's velocity perpendicular to AB is j luxes 
per second, where 

y U-\-ux x , v, 


= V 1 —u 2 X — = Vi— w 2 X— j — 

ux, i-\-uv 


.'. C's velocity perpendicular to AB is 

i luxes per second. 

i-\-uv r 

We see, therefore, that O's measure of C's transverse 
velocity depends on C's longitudinal velocity. 

An interesting illustration of this law of composition of 
velocities is furnished by the motion of light through a 
moving medium. 


FresnePs Convection Coefficient. 

The velocity of light depends on the medium through 
which it is propagated. In a vacuum it moves at 1 lux per 
second, and through air its velocity is nearly as great ; but 
for other media there may be considerable reductions. If 
the refractive index of a medium is /u,, the velocity of light 

through this medium is - luxes per second ; //. of course is 

always greater than 1, for water it is about 1-33. 

Suppose a ray of light is transmitted through a stream 
of water, refractive index fx, which is itself travelling 
through a tube at u luxes per second in the direction of 
motion of the light-ray. With what velocity relative to 
the tube will the light-ray advance ? 

If we regard the ether as stationary, we should expect 
the motion of the water to make no difference to the speed of 
advance of the light-ray. Its velocity in this case would be 

- luxes per second. 

If we regard the water as carrying the ether with it, the 

Newtonian law of composition of velocities would suggest 

that the rate of advance relative to the tube would be 

— \-u luxes per second. 

Experiments by Fizeau in 1851 and by Hoek in 1868 
showed that the actual rate of advance lay between these 

limits and amounted to -+w(i s ) luxes per second. It 

was suggested by Fresnel that this was due to an ether- 
drag ; that, in other words, the ether was only partly 
carried along by the water-stream. 

The Theory of Relativity supplies an alternative explana- 
tion. The light-ray is moving at - luxes per second through 
the water, which is itself advancing at u luxes per second. 


Therefore an outside observer, using the law of addition 
on p. ioo, says that the velocity of the light-ray is 

— \-u 

" luxes per second. 

i + 



1 i ( T i V u \ x , u u 2 
-+« \-+u)ii — ~) -+u — 2 

£ _ > /V - fJ,/ ft ' [A? /J. 

I s 

But u, the velocity of the stream of water in luxes per 
second, is very small. Consequently we shall obtain a 
good approximation if we neglect u 2 . 
In this case, the velocity becomes 

1 i u I , ( i\, 
u'*' u ~u*~u'T u \ L 2/ luxes P er second. 

This agrees with the expression required by experiment. 

We therefore see that the experimental result is a close 

approximation to that deduced by using the Relativity 

formula. The expression i - 2 is called Fresnel's Con- 

vection Coefficient, retails of Fizeau's and Hoek's ex- 
periments will be found in any standard text-book on Light. 

Mass. • 

In the Newtonian System of Mechanics we assign to 
each body a number which measures a property of the 
body called its mass. The mass of a given body is regarded 
as a fixed thing, which is independent of its position or 
velocity, or indeed of any influence brought to bear on it 
so long as no part of the body disappears. Newton said 
that the mass of a body was the quantity of matter it con- 
tained : this phrase is not a definition, but it serves to 
suggest th 3 nature of the concept which the word represents. 


Newton's second law of motion states that the rate of 
change of "the quantity of motion" of a body is pro- 
portional to the impressed force, and this may be said to 
define either his conception of mass or his measure of force. 
From that law we see that if the " quantity of motion " 
of a body is defined by the product of its mass and its 
velocity, the increase in the quantity of motion per unit 
time is the measure of the force acting on the body. 
Instead of the phrase "quantity of motion," the term 
"momentum" is generally employed: 

momentum =mass X velocity. 
This conception of momentum plays a very important 
part in Newtonian mechanics. If any number of bodies 
are in motion, and if there are no external forces acting 
upon the system, then the total momentum of the system 
remains constant. The bodies in the system may exert 
forces on each other, as, for example, by colliding ; but 
such forces are not external and do not affect the sum-total 
of the momentum of the whole system. As the result of 
collisions, etc., there may be a transference of momentum 
from one member to another : the momentum which one 
body gives away is received by one or more other bodies 
of the system. But th£ quantity ot motion or momentum 
of the system taken as a whole never varies unless some 
external force acts on the system and in this way con- 
tributes momentum to the system. Momentum is all 
asset for which an account can be kept. If a number of 
individuals each possess so much money and if this system 
of individuals neither receives money from outside nor pays 
away money to outsiders, but merely engages in internal 
financial transactions, the total capital of the system re- 
mains constant ; any individual member of the society 
can only increase his store of money at the expense of one 
or more other members. It may help the reader if we 
press this analogy still further. 


We are comparing the total capital of a self-contained 
society and the number of members of that society with 
the total quantity of motion (or momentum) of a self- 
contained system of bodies and the number of bodies in 
the system. An observer who looks at the society may 
have a different standard of the value of money from that 
of its members. But his estimate of the total value of 
the capital of the society will remain the same, whatever 
interchanges of cash there are between the individual 
members. His total estimate may not agree with that 
made by another observer who has a different standard of 
values, but, whatever it is, it will remain the same as long 
as no new money is introduced into the system and no 
money is abstracted from the system. Again, the observer 
may see double : in that case he will, when counting the 
number of individuals in the society, obtain a different 
figure from that of an observer with normal vision. But 
as long as there are no births nor deaths, no emigrations 
or immigrations, his census figure will remain the same, 
however the members of the society behave. The observer 
is, in fact, applying a numerical measure to two properties 
of the society, namely, the capital the society possesses 
and the number of members of the society. Both of these 
things are part of the nature of the society, independent 
of any observer, although different observers may apply 
different systems of measurement. But under the condi- 
tions we have enumerated we may say that there is a 
Conservation of Capital and a Conservation of Membership 
of the Society. 

So when dealing with any number of bodies, forming a 
self-contained system, and not subject to any external 
force, we shall say that there is a Conservation of Momentum 
and a Conservation of Mass. The estimate of each of these 
may vary from one observer to another, but whatever 
internal forces may be at work in the system causing 



redistribution of momentum or mass, we say that the esti- 
mates of the total momentum and the total mass made 
by any particular observer will remain constant. 

Now let us suppose that the world of A, B contains two 
bodies of masses m x , m 2 , moving with velocities v lt v 2 luxes 
per second in the direction A->B, and let us suppose that 
an observer O says that A's world is moving away from 
him in the direction A-^B at u luxes per second. 

Fig. 31. 

A says that the total mass of the system is 
m 1 -{-m 2 =c, say, 
and the total momentum is m 1 v 1 -{-m z v 2 =d, say. 

Now c and d are definite constants which will represent 
A's estimate of the total mass and total momentum of the 
system whatever happens to the bodies, given there are 
no external forces. The bodies may collide and so cause 
a transference of momentum ; the collision may break up 
one or both of the bodies into several pieces, etc. In spite 
of all this, taking the system as a whole, A will always 
obtain the same values for c and d : these values, there- 
fore, correspond to some intrinsic property of the system. 

Now if O accepted the principles of Newton's mechanics, 
he would agree with A that the total mass was always 
m 1 +m 2 =c, and he would say the total momentum was 
m^u+Vj) +w 2 (w+v 2 ) 
=m 1 u+m 1 v 1 +m 2 u+m z v z =u{m 1 +m 2 )+m 1 v 1 +m 2 v 2 



O would therefore not agree with A's measure of the 
total momentum, but he would agree that the total mo- 
mentum always remained the same whatever catastrophes 
occurred internally in the system. In other words, if a 
system is behaving so that one observer says that the 
total mass and the total momentum are each remaining 
constant, any other observer would agree with this state- 
ment, although he might disagree with the numerical 
values of the constants. 

But if O accepts the principles of Einstein's mechanics, 
he says that the velocities of the bodies are 

u+v t u+v* . 

f+wi and i+^T 2 luxes P er secon <*> 

and therefore the total momentum is 

i+uv x "•" I+UV 2 
which equals 

\t+uv 1 T.+uv % r\T.+uvi r z+uvj' 
Now although, whatever catastrophes occur, the values 
of m 1 -\-m % and w 1 w 1 +w 2 w 2 remain unchanged, yet subject 
to these two conditions m lt m z , v x , v 2 may vary in value 
in all sorts of ways. And we see that these two conditions 
are no longer sufficient to compel O's expression for the 
momentum to remain unchanged, for we cannot state it 
in terms of u, c, d only. 

In other words, the fact that a system is behaving so 
that one observer says the total mass and the total 
momentum never alter is not sufficient to compel another 
observer to take the same view. 

This means that we must surrender the principle of 
Conservation of Momentum as a property of a system, 
and make it depend on the standpoint of the observer. 
Such a sacrifice would rob mechanics of one of its most 


fundamental principles. Einstein makes this sacrifice 
unnecessary by introducing a new conception of Mass. 

Einstein's Definition of Mass. 

Instead of saying that the mass of a body is independent 
of its velocity, we shall say that if the mass of a body at 
rest in A's world is m, then if the body is moving at v luxes 
per second in A's world, its mass as measured by A is 


Vx—v 2 ' 
It is interesting to note that, long before Einstein intro- 
duced this definition of mass, experimental work had 
suggested that the mass of bodies moving at very high 
velocities varied with the velocity. Sir J. J. Thomson's 
researches on the movements of electrons had shown that 
a high velocity caused an apparent increase of mass of 
amount \mv 2 \ and it can be shown (see Exercise VIII., 
No; 7) that if v is small m+\mv % is a close approximation to 


VT^v 2 ' 

Taking this definition of mass, it is now possible to prove 

that when one observer says that both the mass and the 

momentum of a system remain constant, all other observers 

will agree with him. Momentum is still defined as 

mass X velocity ; therefore, if a body in A's world has a 

mass m when it is at rest, we see that when it is moving 

at v luxes per second in A's world, its mass is 

m . mv 

, — -^ and its momentum is ,- — - 2 . 
vi — v 2 v I— v 

Conservation of Mass and Momentum. 

Using the same notation as on p. 107, A says that the 
total mass is 

/ „ 1 4- , — «- -c, say ; 


and the total momentum is 


Whatever (internal) vicissitudes the system undergoes, A 
will always compute the total mass as c and the total 
momentum as d. 

Let us now examine O's calculations. 

O says that the velocities of the bodies are 

u+v t u+v 2 

■,.„; ■, =r-j — - luxes per second. 

The mass of the first body is therefore 

Now i- ( U+Vl f = ( I +^i) 2 ~(^+ t; i) 2 

_ I + 2UV 1 -f U 2 V X 2 — U 2 — 2UV 1 —V 1 2 
(1+WVi) 2 

_ i— « 8 — p 1 8 +«V _ (i— « 8 )(i— V) 

(i+^j) 2 ~~ (I+Wj) 2 ' 

.'. the mass of the first body is 

m _-. /f(i-« a )(r-Pi a )l «i(i+wi) 

1 " V \ (l+ww x ) 2 J _ Vi_ w axv'i_ l ; 1 2' 

.*. the total mass 

_ m^i+uvj m 2 (i+uv 2 ) 

Vz—u 2 X Vi— v z~t~V-l— u *x Vz—v 2 a 
_ i \ m x -\-um-$) x m^- \-um 2 v 2 \ 

=^=^| c +^}. 


.*. although O assigns a different value to the total mass 
from that assigned by A, as long as A finds that both 
the total mass and the total momentum remain 
constant, we see that O will also find that the total 
mass remains constant. 

O also says that the momentum of the first body is 

m \ u-\-v 1 

f / U J r V 1 \ 2 I+UVi 

Using the results just obtained, we see that this is equal to 

VI^P X Vi — Vj * x YTuvi ° r Vl_ W 2X Vl—vf 
.'. the total momentum 

~ VY^u? x Vi^^ 1 2+ Vi^fi x vr~^ 

- vi=^\ u \ Vi=^ + Vi^r w^v^ •v / i-v 2 V) 

.'. although O assigns a different value to the total 
momentum from that assigned by A, as long as A 
finds that both the total mass and total momentum 
remain constant, we see that O will also find the total 
momentum remains constant. 

Hence, if we accept Einstein's definition of mass, we 
preserve both the Principle of the Conservation of Mass and 
the Principle of the Conservation of Momentum. Both 
mass and momentum are intrinsic properties of the system, 
existing independently of the observers, although different 
observers apply different standards of measurement to 

The above discussion deals only with motion in the direc- 
tion in which A's world is moving away from O. If a body 


at rest in A's world has a mass m, we say that its mass 

, m 

becomes ^— — ^ as computed by A when its velocity is 

v luxes per second in A's world, whatever the direction of 
motion of the body. In order to establish the Conservation 
of Mass and Momentum in the general case, it is of course 
necessary to take account of the formula for the composition 
of transverse velocities ; the method is the same as before, 
but of course the actual algebra is modified. We leave it 
as an exercise for the reader. 

The term " proper mass " is applied to the mass-measure 
of a body at rest in the world in which its mass is computed. 
In the argument used above, m is the proper mass of the 
body in A's world. 

When the body is at rest in A's world, O says that its 

mass is V // I _^ 2 \ - ^ nas Deen pointed out on p. 109 that 
if u is small this is approximately equal to m(i+£w 2 ) or 

Readers acquainted with elementary mechanics will 
recognise that \mu % represents what is called the " kinetic 
energy " of the body— that is to say, the amount of work 
it is capable of doing by virtue of its motion. The term 
" potential energy " is used for the work a body can do by 
virtue of its configuration— for example, a compressed 
spring is said to possess potential energy. The mass of the 
body, which according to O is approximately m+$mu?, is 
therefore equivalent to the sum of its proper mass and its 
kinetic energy. 

The proper mass is therefore of the same nature as the 
kinetic energy, and we may think of this proper mass m 
as representing the potential energy of the body in the 
world in which it is at rest. 

O's measure of the mass of the body is therefore the sum 
of its potential and kinetic energy ; and the fact that the 


mass of a body increases with its velocity is equivalent to 
the statement that an increase of (kinetic) energy shows 
itself by an increase in apparent mass. This leads us to 
identify mass with energy, and to treat the conservation of 
mass as equivalent to the conservation of energy. 

Momentum and Separation. 

The comparative complication of the algebra we have 
used to establish the conservation of mass and momentum 
seems to make the simplicity of the result all the more 
surprising. Was the new definition of mass a happy guess, 
or was there some argument which indicated the form it 
might be expected to take in order to assume an invariant 
character when passing from one world to another ? 

If we replace velocity, which is displacement per unit 
increase of time, by a displacement per unit increase 
of separation, our new type of momentum for uniform 

motion in A's world would be represented by mx- 
where s 2 =* 2 — x 2 . We then have _=i— -=i— v 2 

s t 1 

Vi— v 2 or 

-_ VI _ irWs _ v . 

Hence m x-=m X 7 X - 

1 mv 


Vi — v 2 v'i — v 2 


vi— v 2 
If then m is the proper mass of a body in A's world, we 
may say that the momentum of the body is 

either the proper mass X the displacement per unit 

increase of separation ; 
or the modified mass X the displacement per unit 

increase of time, 


In order to preserve the principle of Conservation of 
Momentum, we could therefore either modify our defini- 
tion of Velocity or, as is done above, modify our definition 
of Mass. 

The Principle of Restricted Relativity. 

Throughout this book we have been continually examin- 
ing various phenomena from the standpoints of different 
observers and attempting to co-ordinate the results re- 
corded by these observers. If any law is enunciated which 
summarises physical processes, it is essential that its 
validity should be recognised by all observers alike. When 
such a law is expressed in mathematical form, it must 
retain that form when we pass from the axes of reference 
adopted by one observer to those adopted by another 
observer. In mathematical language, the form must be 
invariant for any necessary change of axes. We saw that 
the separation between two events was an invariant in 
this sense. Unless a law satisfies this condition, it cannot 
be true. This is Einstein's Principle of Relativity. Hither- 
to we have only considered worlds moving relatively to 
each other with uniform velocity ; and the existence of 
this limitation is indicated by referring to the subject as 
the Restricted Theory of Relativity ; its principles were 
enunciated by Einstein in 1905. The General Theory of 
Relativity takes into account the relations between worlds 
moving relatively to each other with variable velocity ; 
Einstein's investigation of this theory was not published 
till 1915, and owing to the War did not attract attention 
in England till 1917. 

His Principle of Restricted Relativity may be stated as 
follows : Every law of Nature which holds good with respect 
to one co-ordinate system (say A's world) must also hold 
good for any other co-ordinate system (say O's world), 


provided that A's world and O's world are moving with 
uniform velocity relatively to one another. 

This is equivalent to the statement that it is impossible 
to devise any experiment which will detect uniform motion 
through the ether. For, if a law was valid hi one world 
only, it would indicate something unique in the nature of 
that world, and enable that world to be taken as a standard 
of reference. And equally if uniform motion through the 
ether could be detected, we should thereby possess a 
unique criterion for distinguishing one special world from 
all the others. 

The purpose of the Theory of Relativity is to distinguish 
the subjective impressions of the observer and the relations 
which connect him with what he observes from the reality 
of the thing observed. The property of separation occupies 
a central position because it is a property of physical reality. 
The name " Relativity " is to some extent misleading, as 
it tends to imply that physical inquiry is of a relative nature. 
What Einstein has set himself to achieve is a formulation 
of physical principles which are independent of the observer. 
His work may, therefore, more justly be called a Theory of 
Physical Reality, or a Theory of Space-Time Events. 


i. A body C is moving in the world of A along AB with a 
velocity £ lux per second ; O says that A's world is 
moving in the direction A-kB at (i) £, (ii) & lux per 
second. What is C's velocity according to O ? 

2. Repeat No. i, using the same data, except that C is moving 

in A's world in the direction B->A. 

3. A body C is moving in the world of A at £ lux per second 

at right angles to AB ; O says that A's world is moving 
in the direction A->B at f lux per second. What is C's 
velocity according to O ? 

4. A body C at rest in A's world has a mass 2 as measured by A. 

If C now moves at § lux per second along AB in A's 


world, what is A's measure of its mass ? If O says that 
A's world is moving along AB at } lux per second, what 
is O's measure of its mass ? 

5. If C is in motion with the data of No. 4, evaluate C's momen- 

tum as measured (i) by A, (ii) by O. 

6. A body, whose proper mass is 5, is moving at o-i lux per 

second ; show that its apparent increase of mass is 
approximately equal to its kinetic energy (half mass x 
velocity 2 ). 

7. Show that (1 +iw 2 ) 2 (i — 1/8) = 1 _f w «— Jt/«. Hence if v* is 
that 1+^v 2 is a close approximation to 


" I have made such wonderful discoveries that I am myself 
lost in astonishment : out of nothing I have created a new and 
another world." — John Bolyai ; a letter to his father on 
non-Euclidean Geometry, dated 3rd November 1823. 

In the restricted Theory of Relativity, we have considered 
only a special class of observers, namely, those who are 
moving relatively to the events observed with uniform 
velocity. In this theory we have seen that there is no 
justification for selecting any special observer as a Court 
of Appeal. It now becomes necessary to proceed to a 
more general inquiry. What differences arise if observers 
and events move relatively to each other with variable 
velocity ? Motion of this kind is of very common occur- 
rence ; its investigation forms what is called the general 
Theory of Relativity. 

Force and Acceleration. 

If a stone is dropped from the top of a tower, it falls 
with a velocity which increases with the time. If we 
neglect air-resistance, the velocity increases at a constant 
rate : after 1 second its velocity is 32 feet per second, after 
2 seconds its velocity is 64 feet per second, after 3 seconds 
its velocity is 96 feet per second, and so on. We say that 
the body is moving with uniform acceleration, and that its 
measure is 32 feet per second every second. The accelera- 
tion of a body is the increase of its velocity per unit time. 
If the velocity is diminishing, the acceleration is negative. 


According to Newtonian mechanics, a body is accelerated 
(i.e. its velocity is changing in magnitude or direction or 
in both ways) */ and only if a force is acting upon it ; and 
the magnitude of the force is represented by (i) the increase 
of momentum per unit time, or in other words (ii) mass 
X increase of velocity per unit time, or in other words 
(iii) mass x acceleration. 

If a body A pushes against a body B, it is easy to picture 
the nature of the force which A exerts on B. We can 
think of B as being struck by a very large number of 
molecules belonging to A, moving at very high speed and 
thereby transferring some of their own momentum to the 
molecules of B. Force exerted by what appears to be 
actual contact therefore seems an intelligible operation, 
and forces of this kind play a large part in ordinary life. 
The seat of the chair in which I am sitting is bombarding 
me with molecules, all giving me some of their momentum, 
and producing in me a consciousness of force which I call 
the pressure exerted by the chair. But this momentum 
I am receiving is spent as rapidly as it is given. What is 
the cause of the expenditure ? According to Newton, it 
is a force of attraction towards the centre of the Earth 
which the Earth exerts on me — the force of gravitation. 

Clearly this is a force of an entirely different character ; 
it does not act by means of direct contact, but appears 
as an intangible influence radiating throughout space and 
diminishing inversely as the square of the distance from 
the source. Mention has already been made (see p. 6) 
of the difficulty, which familiarity has disguised, of accept- 
ing an hypothesis involving " Action at a Distance." But 
there is an even more remarkable property peculiar to 
gravitational forces. 

Arrange an experiment so that two bodies are free to 
move under the gravitational attraction of the Earth, 
with no other forces influencing the motion. Newton 


took a closed cylinder in which he placed a feather and a 

guinea, and then exhausted the air from the cylinder. The 

two bodies were then allowed to fall simultaneously from 

the top of the cylinder. They fell side by side and struck 

the bottom at the same moment. Apart from the action 

of other forces, all bodies, however different in size, shape, 

or mass, fall with equal accelerations under the influence 

of gravity, provided they start from the same place. In 

other words, the acceleration of a body due to gravity 

depends solely on its position in space and is independent 

of size, shape, composition, and mass of the body. The 

acceleration is a function of position : near the Earth's 

surface we know by experiment that the acceleration is 

approximately 32 feet per second every second. Outside 

the Earth, the acceleration varies inversely as the square 

of the distance from the centre of the Earth. Taking the 

Earth's radius as 4000 miles, it follows that 8000 miles 

away from the Earth's centre the acceleration is y 2 = 8 feet 

per second every second, and at 12,000 miles from the 

Earth's centre it is ^=^=3'6 feet per second every 

second, and so on. If we limit our attention for the 
moment to bodies not more than a mile or two above the 
Earth's surface and within a mile or two of each other, 
we may regard the gravitational acceleration as practi- 
cally uniform, and we shall say that the bodies are situated 
in a uniform field of force. 

Fields of Force. 

If an observer is watching a body in motion and notes 
that it is moving in a curve, he will say that there must be 
some force acting on it. This is required by Newton's 

Suppose, however, some other onlooker, watching the 


same series of events, says that the body is moving uni- 
formly in a straight line and therefore is not subject to 
any force. Who is right ? Is it possible for two equally 
honest and competent observers to differ as to whether 
the path of the body is straight or curved ? Is it possible 
for the statements that (i) a force acts on the body, (ii) no 
force acts on the body, to be equally true ? Is it possible 
to say that one observer by virtue of his circumstances is 
better qualified than the other to judge what is happening ? 

Suppose A lives in a large transparent airtight glass box, 
say the size of Olympia. A's home is taken to a place 
several thousand feet above the Earth's surface and allowed 
to fall. B stands on the ground and watches A's behaviour 
through a powerful telescope. For the sake of simplicity 
we shall suppose there is no air-resistance. 

B's observations will be based on the fact that B regards 
himself as at rest on the Earth's surface and regards A as 
falling vertically with an acceleration of 32 feet per second 
every second. Everything in A's house is behaving in 
the same way — the pictures on the walls, the pipe in A's 
mouth, a tennis ball which A holds in one hand and a spring- 
balance which he holds in the other. When A throws the 
ball across the room, B says that the ball describes in space 
a curve called a parabola, the curve in which any projectile 

Now consider A's sensations. If A wishes to decorate 
a wall of his room with a picture, he holds it up against 
the wall and leaves it there ; there is no need to suspend 
it from a hook ; when he takes his hand away, the picture 
remains where he has put it. B, of course, says that the 
wall and the picture both fall at the same rate. A takes 
his pipe out of his mouth and drops it, but the pipe remains 
stationary. B says that both the pipe and A fall side by 
side. When A throws the ball across the room, he judges 
that it moves in a straight line until it collides with some 


object in the room) although B says its path is curved. A 
attaches a table to the spring-balance and notes that its 
weight is zero ; he then stands on the platform of a weigh- 
ing-machine and observes that his own weight is zero. 
All these experiments convince A that he is at rest in a 
space free of gravitational attraction, while B is equally 
convinced that A is falling in a uniform field of force. 
Previous discussions in this book make it easy to reconcile 
the discrepancy between the opinions of A and B. Whereas 
B chooses axes of reference attached to the Earth, A takes 
axes of reference attached to the glass box. These two 
systems are moving relatively to each other with uniform 
acceleration ; and the consequence of this is that a path 
A calls straight, B says is curved, and a region which A 
declares is free of force, B says is a uniform field of force. 
The relation is reciprocal : A says that the Earth and B 
are falling towards him with a uniform acceleration. If 
B rolls a ball across a level table, A says that the ball 
actually describes a parabola in space, and so on. 

The Principle of Equivalence. 

Our natural inclination is to associate ourselves with 
B's view rather than A's, but this is a parochial attitude, 
and is due to the fact that we are accustomed to live under 
the same conditions as B does. If, however, we were a 
race of falling aviators we should sympathise with A 
rather than B. The Theory of Relativity forbids us to prefer 
any one observer to any other ; there is to be no favouritism 
or prejudice : any law of Nature must be equally acceptable 
to all observers, and must therefore take an invariant form 
which survives transformation from one world to another. 
Gravitational force is an illusion. This does not, of course, 
mean that if you throw yourself off the top of a tower, 
trouble will not ensue. But Einstein denies that the 
event consisting in your subsequent collision with the 


ground is caused by the Earth exerting an attractive force 
on you. We shall soon be in a position to give Einstein's 
explanation of the course of events. 

So far we have only considered the effect of gravitation 
over a small region throughout which the Newtonian 
theory regards it as setting up a uniform field of force. 
We have seen that in this case the effects can all be removed 
by a change of axes. The existence of a uniform field of 
force as affirmed by B is denied by A, who chooses axes 
moving with uniform acceleration relative to B's.axes. 
Another observer with a different system of axes would 
affirm the existence of a different field of force. Conse- 
quently we may say that these fields of force are imputed 
by the observer to the Universe owing to his own local 
circumstances. A suitable change of axes will neutralise 
any uniform field of force. It therefore follows that a 
uniform field of force is artificial, an unconscious invention 
of the observer, rather than a property of the thing observed. 

A has by his choice of axes neutralised in his own neigh- 
bourhood what B calls a gravitational field, but in so doing 
he has made B appear to be falling toward him with an 
acceleration 32 feet per second every second, and if he can 
see through the Earth and observe another aviator C 
crashing there, he will impute to C an acceleration of 64 feet 
per second every second. A therefore says that B is in a 
field of force and C in another field of double the intensity : 
similarly with other falling aviators elsewhere, A will 
impute to each a field of force of different intensity and 
direction. Consequently, although A's choice of axes re- 
moves the effect of gravitation in his own neighbourhood, 
it makes matters worse elsewhere, by creating fields of 
force of all sorts of various magnitudes and directions. 
However, that does not perhaps matter to A. The con- 
clusion of these remarks forms Einstein's Principle of 
Equivalence : 4 


If attention is confined to a small region of space, a gravita- 
tional field at rest is equivalent to a frame of reference moving 
with uniform acceleration in a field free of gravitation ; and 
it is impossible to devise any experiment which will distinguish 
between the two. 

We therefore see that although the presence of matter 
is responsible for creating a gravitational field, yet any 
observer, just like the inhabitant of the glass box, can so 
choose his axes that in his immediate neighbourhood all 
gravitational effects are neutralised, and consequently, 
within this small region, although not beyond it, the 
principles of the theory of restricted Relativity apply. This 
conclusion is of the utmost importance, because it enables 
us to use within these limits results established for space- 
time uninfluenced by matter. 

Space-Time Distortion. 

Let us now consider the world-line of a body moving 
through a space-time domain, in which matter is present, 
and let us suppose that an observer is moving with, in fact 
travelling on, the body. At each point in space-time, the 
observer by his choice of axes can and does neutralise the 
gravitational field in his immediate neighbourhood. He 
can use his own clock to measure the separation of two 
events in his career if these events are very close together ; 
the separation will be simply the proper time as recorded 
by his clock. By a process of summation he will then find 
the total measure of the separation between any two events 
in his career, measured along his world-line. If the body 
is moving freely, the observer moving with it says that the 
world-line is straight, and consequently that the path in 
space-time is such that the separation between two events 
is a maximum. If then the measure of the separation is 
the same for all observers, the path must be such that every 
other observer will find that the separation measured 


along it is a maximum. But other observers will not say 
that the world-line is straight ; from their points of view the 
principles of the restricted theory do not apply. The fact 
that the geodesic, i.e. the route for maximum separation, 
is a straight line depends on the restricted theory where 
space-time is uniform. This is no longer true. 

The other observers say that the presence of matter has 
distorted space-time in its neighbourhood, and as a result 
of this the geodesic is a curve. According to Newton, the 
Earth describes an ellipse round the Sun owing to an 
attractive force which the Sun exerts on the Earth. But 
according to Einstein the presence of the Sun causes irregu- 
larities in the space-time in its neighbourhood, and the 
Earth simply picks its way through this tangled domain 
following a path (can we call it a spiral ellipse in space- 
time ?) so devised that when we allow for the crumpling 
up of space-time the separation measured along it between 
any two given events is a maximum. In other words, the 
Earth's orbit is curved not because the Sun exerts any force 
on it, but because in the distorted space-time domain round 
the Sun the geodesic is not straight but curved : it is easier 
to move through the obstacles by following a curved route, 
just as in passing through a wood in which the trees are 
denser in some parts than others it is often easier to follow 
a curved route than to try to go straight ahead all the time. 

This conception of distorted space may be easier to 
appreciate by considering another illustration of a field of 

Life on a Rotating Disc. 

Imagine a large plane disc, centre C, which an outside 
observer O says is rotating about an axis through C per- 
pendicular to its plane. Another observer A lives on the 
disc and draws through C axes of reference on the disc 
along and perpendicular to CA. 


A regards the disc as at rest, and thinks that O is moving 
in a circle in the reverse direction. But A realises that he 
has to attach himself to the disc 
in order to keep his footing. 
A believes that there is a gravi- 
tational field of force acting . . . ( . x 
outwards from C and propor- \ \^ c \J A J O 
tional to the distance from C. 
But O says that A is travel- 
ling round C in a circle with Fig. 32. 
uniform speed ; and therefore 

has an acceleration towards C which is produced by A 
holding on to the disc, just as a stone attached to the 
end of a string and whirled round in a circle is held to its 
circular path by the pull of the string. Suppose now a 
body starts from C and moves with uniform velocity along 
CO towards O. Then of course O says that it is travelling 
in a straight line in a field devoid of force. How will A 
view the progress of the body ? A thinks that the disc is 
at rest and that it is O who is revolving. Consequently 
A will say that the body travels outwards from C along the 
line CO, which is itself rotating. A therefore, tracing the 
position of the body relative to his axes on the disc, says 
that the body describes a sort of spiral curve ; and natur- 
ally he attributes the curved orbit to the gravitational 
field which he believes is existing. What therefore O 
regards as a straight path in a field of no force, A regards 
as a curved path in a gravitational field. 

Suppose now a circle is drawn on the disc with C as centre, 
and that A uses his rule to measure (i) its diameter, (ii) its 
circumference. Suppose that A finds the diameter is equal 
to 1,000,000 lengths of his rule, O will agree with this 
measurement, because in any radial position the rule has 
no velocity in the direction of its length, relative to O. But 
when A places the rule tangentially to the circle and 


proceeds to measure the circumference by stepping it off 
in small bits, the rule has a velocity in the direction of its 
length relative to 0, and therefore says that the rule 
contracts. O knows that the circumference of the circle 
is equal to 7rX diameter, where ^=3-14159265 . . ., and 
therefore 3,141,592 steps would be required to traverse the 
circumference if the rule did not contract, but owing to 
the contraction more steps will be necessary ; the number 
of course depends on the contraction ratio. O watches 
A perform the process, and notes that it takes (say) 3,300,000 
steps, using the contracted rule. O and A must of course 
agree in any counting process. A is surprised by this 
result, because he is unconscious of any contraction of the 
rule, and is forced to believe that the ratio of the circum- 
ference to the diameter is no longer 3-14159 . . ., but is in 
this case 3-3. 

A now repeats the process with a larger concentric 
circle ; suppose its diameter is double that of the first 
circle. O and A then agree that the diameter is 2,000,000 
steps of the rule. O says that the speed of A is now twice 
what it was before, and therefore the contraction-ratio 
is greater than before, and so it now takes 8,000,000 steps 
of the rule to measure round the circumference (see Exer- 
cise IX., No. 7). A is therefore compelled to say that the 

ratio of the circumference to the diameter is ~— 

2,000,000 ^ 

We see, then, that in A's world the circumference of a 
circle is not proportional to the diameter ; in other words, 
two circles of different size are not similar (i.e. are not the 
same shape). A's geometry does not therefore agree with 
the geometry of Euclid, and we say that A's space is 

But there is also another curious feature of A's world. 
Since the velocity of A relative to O increases proportionally 
to the distance of A from the centre C, O says that the 


clocks in A's world do not run at the same rate ; the 
farther a clock is from C, the slower it runs according to 
O. Time-measure, as judged by O, is therefore not uni- 
form in A's world. In the restricted theory we saw that 
said A failed to synchronise his clocks and that all the 
clocks ran slow, but O admitted that they all ran at the 
same rate : time was uniform all over A's world, although 
its measure was different from that used by O. Here, 
however, there is a new element of irregularity, for^A's 
clocks run at a rate which depends on their distance from C. 

The irregularity of space-measurement is also accom- 
panied by an irregularity of time-measurement. The 
space-time world of A is distorted, both in respect of time 
and in respect of space. 

O, of course, considers that both space and time are 
uniform : the non-Euclidean character of A's space and 
the irregularity of time are due to A's creation of a gravita- 
tional field, arising from his choice of axes. O's choice of 
axes has made the space-time domain uniform ; A's choice 
of axes is equivalent to setting up a gravitational field 
which shows itself in the distortion of space and time. 

Gravitational Fields. 

The existence of matter gives rise to a gravitational 
field in its neighbourhood ; but instead of saying that this 
is a field of force, we now say that it is a distortion of space- 
time. The invariant expression for the separation between 
two events, s 2 =t 2 — x 2 — y 2 — z 2 , was established by assuming 
that space-time is uniform : since in the neighbourhood 
of matter this is no longer the case, this expression will 
require modification in a gravitational field, if it is to sur- 
vive transformation from one world to another. A new 
geometry is automatically imposed upon us, with a different 
set of mensuration formulae. Various systems of geometry 
have been investigated during the last hundred years. 


Up till that time it had been assumed that the geometry 
of Euclid was the only possible logical system. The 
characteristic property of Euclid's geometry is that the 
sum of the angles of a triangle is two right angles. It is 
now universally accepted that equally consistent Geometries 
exist which conflict with Euclid. In what is called Hyper- 
bolic Geometry the angle sum of every triangle is less than 
two right angles, and in Elliptic Geometry the angle sum 
is always greater than two right angles. If we ask which 
of these is really true, the question can only mean, which 
of these applies to the world in which we live. Gauss 
attempted to answer the question by taking a large triangle 
whose corners were the summits of three mountains and 
measuring the angles. But the difference of their sum 
from two right angles was less than probable experimental 
errors. There is no doubt that only a triangle whose 
sides involve lengths of astronomical magnitude can give 
a decisive answer to an inquiry conducted in this way. 
Gauss's experiment could not possibly lead to a decision. 
Such evidence as exists at the present time points to the 
theory that the geometry of our Universe is Elliptic, and 
this involves the supposition that it is finite in extent, 
finite but unbounded, just as the surface of a sphere is 
finite but without a boundary. 

The educated man of to-day understands the general 
characteristics of the Newtonian theory, but only the 
specialist can read and understand the Principia in which 
the formal investigation is made. In the same way the 
mathematical process used by Einstein to deduce the laws 
of the space-time geometry of our Universe and the formu- 
lation of those laws in all their generality can only be ap- 
preciated and apprehended by the mathematical specialist. 
But the nature of the ideas which distinguish Einstein's 
theory from Newton's can be illustrated without any 
advanced mathematical reasoning, and it is possible to 


state in a simple form Einstein's law of gravitation for the 
special case which affects us most, namely, for the portion 
of space-time round about the Sun. 

Suppose that S is the centre of a massive body such as 
the Sun, and suppose that P and Q are two events near 
together in space and time. From 
Q draw the perpendicular QN to SP, 
produced if necessary. Since Q is 
near P, we regard QN and PN as 
small compared with SP, and the 
length of each is to be measured 
in luxes. Suppose also that the 
time-interval from P to Q is t 
seconds where t is also small. Then 
if m is the gravitational mass of the massive body at S, 
also measured in luxes, the separation of Q from P is 
given by 

*-(i-»y_ 0N ._( I+ |») . p N *. 

The method for calculating m is shown in Exercise IX., 
Nos. 5, 6. For the Sun, m =0-000,004,9, and for tne Earth 
m =0-000,000,000,02. It should be noted that if m=o, this 
expression becomes s 2 =t 2 — QN 2 — PN 2 =* 2 — PQ 2 , which is 
the ordinary form for the separation in the restricted theory : 
putting m equal to zero is of course equivalent to saying 
that no matter is present to influence the nature of the 
space-time in which the events occur. The introduction 
of the additional terms involving m therefore represent 
the changes in the mensuration formulae caused by the 
distortion of the space-time domain surrounding a single 
massive body. 

If a body is moving freely in the neighbourhood of a 
massive body, it follows a path so chosen that the separa- 
tion, as detennined by the formula given above and 
measured along this path, is a maximum. Newton's Law 


of Gravitation is replaced by Einstein's geometrical men- 
suration formula for a space-time domain, a geometrical 
space-time Law. Einstein's Law may be tested by 
examining the paths in which the planets move round the 
Sun. Are these paths geodesies in space-time, when the 
separation is calculated according to the formula given 
above ? That the paths of the planets fit very closely 
with the paths as calculated in the Newtonian theory is 
well known. It is simply because Newton's Law of 
Gravitation leads to orbits which agree so closely with the 
observed orbits that up till the time of Einstein this law 
was universally accepted. But we shall see in the next 
chapter that close as is the agreement between calculation 
and observation, a still higher degree of accuracy is secured 
by the substitution of Einstein's Geometrical Law for 
Newton's Mechanical Law : the latter may indeed be 
regarded as a first approximation towards the former. 


i. A toy pistol is pointed straight at the bull's-eye of a target. 
At the moment the pellet leaves the pistol, the target is 
allowed to fall vertically. Will the pellet strike the 
bull's-eye, if air resistance is ignored ? Compare the 
views of the path traced out by the pellet formed by 
(i) the boy who holds the pistol, (ii) a microbe on the 

2. Suppose that A, whose weight is 10 stone when on the 

ground, is standing on a weighing-machine in his glass 
box (p. 120), and notices that his weight has changed 
from zero (i) to 10 stone, (ii) to 20 stone, (iii) to 100 stone, 
what will he say about gravitation ? How will B, who 
is standing on the ground, account for it ? 

3. Three people are watching a body. One says it is at rest, 

the second says it is moving in a straight line, and the 
third says it is moving in a curve. Is it possible that 
all three observers are equally efficient ? 

4. Suppose that the disc on p. 125 is rotating at the rate of 


5 revolutions per minute, and that a body is moving at 
a uniform rate of 5 feet per minute just above the disc 
along the straight line CO. Plot its positions at intervals 
of 1 second for 12 seconds, as recorded by A, who uses 
lines marked on the disc as axes of reference. 
If the gravitational mass of the Sun is m, the acceleration 
towards the Sun of a planet at a distance of r luxes from 

the centre of the Sun is approximately ~^. If the planet 

is moving in its orbit at v luxes per second, it is known 

that the acceleration radially inwards is — . Hence 

fyt v^ 

-a=— or m=v*r. Taking the distance of the Earth from 

the Sun as 500 luxes, use this formula to show that the 
gravitational mass of the Sun is about 1-5 km., and 
express the result in luxes. 

Taking the distance of the Moon from the Earth as 240,000 
miles and the period of a revolution as zj\ days, use the 
formula in No. 5 to show that the gravitational mass of 
the Earth is about 5 millimetres, and express the result 
also in luxes. [1 lux = 186,000 miles.] 

In the measurement of the circumferences of the two circles 
on the rotating disc on p. 1 26, show according to O that 

(i) the first speed of A is given by i — u a = (^^J , (ii) the 

contraction-ratio along the larger circle is about 079, 
(iii) the number of steps of the rule round the larger 
circle is about 7,970,000. 


" In one sense deductive theory is the enemy of experi- 
mental physics. The latter is always striving to settle by 
crucial tests the nature of the fundamental things : the former 
strives to minimise the successes obtained by showing how 
wide a nature of things is compatible with all experimental 
results. — A. S. Eddington, Mathematical Theory of Relativity. 

A scientific theory maintains its position only so long as 
it harmonises observed facts. If discrepancies between 
theory and observation remain, after full allowance has 
been made for possible observational errors, then modifica- 
tions must be made in the theory. 

The Perihelion of Mercury. 

For a long time it had been realised that there was a 
serious difference between the observed orbit of Mercury 
and the path obtained by calculations based on Newton's 
Law of Gravitation. If Mercury were the only planet in 
the solar system, its path would be an oval curve called 
an ellipse having the Sun at a point inside it known as the 
focus S ; the centre C of the curve 
is a different point ; if CS cuts the /^ ~^^ 

orbit at A, A' as shown, then the Ar— 1--+ JA 

planet is nearest the Sun when at \ y 

A and is farthest from the Sun when p IG 34 

at A'. The point A is called the 

■perihelion of the orbit. Owing to the attraction exerted by 

the other planets, using Newtonian language, the regularity 


of motion is upset, and instead of moving in exactly the same 
elliptic orbit, year after year, the path is represented by 
an ellipse in which the perihelion is steadily advancing, i.e. 
SA is steadily rotating relatively to the fixed stars. Calcula- 
tions reposing on Newton's Law of Gravitation show that 
the combined influence of all the known planets would 
cause a rotation of 532 seconds of angle per century. 
Observation, however, shows that it is actually rotating 
at the rate of 574 seconds per century. There is con- 
sequently a discrepancy of 42 seconds per century to be 
accounted for. This may sound a very small error — it is 
less than the angle which a halfpenny subtends at the eye 
from a distance of 135 yards— but actually it is far greater 
than any possible observational error. 

The irregularities in the motion of Uranus were respon- 
sible for the dramatic discovery of Neptune. In 1846, 
Adams and Le Verrier, working independently, calculated 
the path of a planet which would produce the observed 
perturbations in the orbit of Uranus. The calculated 
position in the sky of this hypothetical planet was sent by 
Le Verrier to Dr. Galle of Berlin, who at once turned his 
telescope to the place indicated in the sky and discovered 
the new planet Neptune very close to the position predicted 
for it. An attempt was made to account for the irregularity 
in Mercury's motion in a similar fashion : and a planet 
to which the name Vulcan was given was invented for the 
purpose. But Vulcan has never been found, and its 
existence is now discredited. If, however, we substitute 
Einstein's Law of Space-time for Newton's Law of Gravita- 
tion, the discrepancy disappears. The deduction of this 
result from Einstein's Law requires mathematics of too 
advanced a character for these pages, but the additional 
correction which Einstein's Law supplies can be stated in 
simple language : if a planet describes its orbit with a speed 
of v luxes per second, the line joining the Sun to the 


perihelion rotates through an additional angle of amount 
I2v 2 right angles per revolution. The reader can easily 
verify (see Exercise X., No. 3) that in the case of Mercury 
this correction amounts to within a second of the needed 
42 seconds per century. 

It is very unfortunate that it is impossible to check this 
correction by reference to any of the other planets. For 
in the case of every other planet either the speed is too 
small or else the orbit is so nearly circular that accurate 
observation of the position of the perihelion is impossible. 
But the fact that it gives so accurately the necessary 
correction for Mercury's motion is a powerful and striking 
argument in favour of the Relativity theory. 

In addition to showing how his theory removed an 
anomaly of which no satisfactory explanation had pre- 
viously been given, Einstein made two predictions which, 
if capable of being submitted to an experimental test, 
would serve to distinguish between the old theory and 
the new. 

Shift of Spectral Lines. 

The vibration of an atom may be regarded as supplying 
us with an ideal natural clock. If two atoms are identical, 
and if we measure the separation between the beginning 
and end of a vibration, the result should be the same 
wherever the atoms are situated. Now suppose one of 
the atoms is close to the surface of the Sun and that the 
other is in a laboratory on the Earth. We may regard 
the events for each atom as happening at the same 


Suppose that the period of vibration of the solar atom 
is t x seconds, and for the terrestrial atom is t 2 seconds, 

/ 2W\ 

then, for the solar atom, s a =^i — gp^i 


where SP=43o,ooo miles=2-3 luxes, ^=0-000,004,9 luxes. 
And for the terrestrial atom, s 2 =(i-|^V 2 2 

by 7 
where SQ =93,000,000 miles =500 luxes, 
m =0-000,004,9 hixes. 
The value of s is the same in each case 

• • v I— i^Ai =V I— 5o^M where m =0-000,004,9. 
It is, therefore, clear that t x is a little greater than t 2 . 
The reader may see for himself that j is approximately 

1-000,002. Consequently the solar atom vibrates just a 
little more slowly than the atom on the Earth. Now the 
time of vibration affects the colour, and therefore in the 
solar spectrum there should be a very small shift towards 
the red end of the spectrum as compared with the spectrum 
of the same atom on the Earth. 

The shift is, however, in general so minute as to defy 
measurement. But Professor Eddington pointed out some 
time ago a case where a larger shift might be expected. 
There is a companion star of Sirius, known as a " white 
dwarf," i.e. an early type of star of very low intrinsic 
brightness. It is believed that the density of this star 
(the statement sounds incredible !) is more than 30,000 
times the density of water — that is to say, its mass is more 
than half a ton per cubic inch ; its radius is about 12,000 
miles. Consequently it sets up a gravitational field of so 
great an intensity in its neighbourhood that a measurable 
shift is likely to occur. Mr. Adams of Mount Wilson 
Observatory has recently obtained results for the displace- 
ment of some of the hydrogen lines of the spectrum of 
this star, which appear to confirm the Einstein prediction, 
although the method involved considerable difficulties 
both of observation and measurement. 


Curvature of Light-Rays. 

Newton envisaged the possibility that light has weight. 
It is now an accepted fact that a ray of light does exert 
radiation pressure on any object on which it impinges : 
this is equivalent to saying that light possesses mass. 
Naturally the quantities involved are small : Professor 
Eddington, in his volume Space, Time, and Gravitation, 
states that the mass of the total amount of sunlight im- 
pinging on the Earth every twenty-four hours is about 
160 tons. But if light has mass, whether we follow Newton's 
or Einstein's Law, a ray of light passing near the Sun 
should move in a curve in just the same way as do the 
planets or comets. The fact that light moves so much 
faster than any planet or comet will naturally mean that 
the amount of the deflection when passing near the Sun 
is very much less. 

If, then, a ray of light travel- 
ling from a star P towards the „*' _ 
Earth E passes near the Sun S 
and is slightly deflected so that 
ES is not in the same straight 
line with SP, then the star, 
as seen from the Earth, will Fig. 35. 
appear to be in the direction 

ESQ, whereas its true direction is the line EP. The 
angular displacement between the true position and the 
apparent position of the star in the sky is represented 
by the angle PEQ. This angular displacement can be 
calculated, but different results are obtained according as 
we adopt Newton's or Einstein's Law. It has already been 
pointed out that Newton's Law may be regarded as a 
first approximation to Einstein's Law. The latter, so to 
speak, adds on to Newton's Law a correction arising from 
the] distortion of space in the neighbourhood of matter. 


If we were to replace the formula on p. 129 by the relation 

i~Sp,F-QN 2 -PN 2 , we should obtain orbits which 

agree closely with those deduced from Newtonian principles. 

The addition of the term — gp . PN 2 corresponds to a 

substitution of non-Euclidean for Euclidean space (not 
space-time), and it is the presence of this term which makes 
a decisive difference between the calculated value of the 
apparent displacement of a star in the two theories. It is 
impossible to reproduce the necessary calculations in these 
pages, but the result can be stated in a simple form. If a 
ray of light from a star passes the Sun at a distance r luxes 
from the centre of the Sun, then, as viewed from the Earth, 
the angular displacement according to the Einstein theory 

is — right angles, where as before m =0-000,004, 9 and 
^=3-14. If the ray passes close to the Sun's surface we 
may take 7=700,000 km. =2-3 luxes. We leave it to the 
reader (Exercise X., No. 4) to show that this is equivalent 
to an angle of about 175 seconds. The Newtonian theory, 

on the other hand, leads to a displacement of — right 
angles, just half the amount required by Einstein. It 
lies with the astronomer by actual observation to judge 
which is correct. 

Sir Oliver Lodge, in the course of an article in the Nine- 
teenth Century, gave a vivid illustration of the task this 
test imposes on the practical astronomer ; part of it we 
venture to quote : 

" Take a fine silk thread of indefinite length and stretch 
it straight over the surface of a smooth table. Imagine a 
star at one end of the thread and an eye at the other end, 
and let the thread typify one of the rays of light emitted 
from the star. Now take a halfpenny and place it on the 
table close to the thread so that the eye end of the thread 


is 10 feet away ; and then push the halfpenny gently 
forward till it has displaced the thread the barely per- 
ceptible amount of T ^ inch. The eye, looking along the 
thread, will now see that the ray is no longer absolutely 
straight ; in other words, the star whose apparent position 
is determined by that ray will appear slightly shifted. The 
scale is fixed by the size of the halfpenny, whose diameter, 
1 inch, is used to represent the Sun's diameter of 800,000 
miles. The 10-foot distance between eye and Sun practi- 
cally supposes the eye is on the Earth, which would be a 
spot about the size of this full stop. As for the distance 
of the star at the far end of the thread, that does not 
matter in the least ; but on the same scale for one of the 
nearest stars the thread would have to be about a thousand 
miles long. The shift of ^Vjt inch at a distance of 10 feet 
corresponds to an angle of if seconds, which is just the 
optical shift that ought to occur, according to Einstein, 
when a ray from the star nearly grazes the Sun's limb on 
its way to a telescope." 

Modern apparatus and methods have now attained 
such a high degree of refinement that the measurement 
of even so small an angular displacement as is indicated 
by this illustration, or rather the discrimination between 
two small angular displacements of this order of magnitude, 
is well within the powers of the present-day astronomer. 
Unfortunately the only time when a star nearly in line with 
the Sun is visible is during a total solar eclipse, and, in 
addition to this, clear observations are difficult to obtain 
unless there happen to be at the time of the eclipse several 
bright stars in the direction under survey. By good 
fortune, a total solar eclipse occurred on May 29, 1919, 
when these conditions were satisfied. Two expeditions 
were organised, one being sent to Sobral, in North Brazil, 
and the other to Principe, in the Gulf of Guinea, to take 
the necessary photographs. The story of these expedi- 


tions has been told in detail by Professor Eddington (see 
Space, Time, and Gravitation, ch. vii.) ; the expedition to 
Principe fared badly, because clouds interfered seriously 
with the operations, but at Sobral the atmospheric condi- 
tions were excellent. Here, however, there were other 
complications which diminished the value of many of 
the photographs. Allowing for probable experimental 
errors, the Principe observations gave an apparent dis- 
placement of between 1-91 and 1-31 seconds, while the Sobral 
observations gave a displacement between 2- 10 and 1-86 
seconds. Another total eclipse occurred in 1922, and a 
successful expedition was organised by the Lick Observa- 
tory. The published results of one set of photographs 
{Lick Observatory Bulletin, No. 346) give the mean value 
as 172 seconds (or if a certain correction is applied as 
2-05 seconds) with a possible error in defect or excess of 
about 0-12 seconds. It is believed that the records of 
another set of photographs, not yet published, are even 
more favourable to the Einstein deflection. There will 
be another very favourable solar eclipse in 1938. 

Many readers may feel disappointed that there should 
appear to be so wide a margin in the recorded results. 
It is necessary to engage in practical work to appreciate 
that sources of experimental error are inevitable : all that 
can be done is to indicate the probable margin of error. 
It is also necessary to remember that the conditions under 
which an expedition has to work are far less favourable 
than those of a permanent observatory, just as a field 
telephone company on active service is far less favourably 
placed than a London Telephone Exchange. But the 
records undoubtedly support Einstein as opposed to 
Newton. If it is necessary to choose between the two, 
there can be no doubt that the Einstein calculations are 
in closer agreement with eclipse observations than those 
based on the Newtonian theory. But apart from the con- 


crete evidence furnished by observation, it is important to 
remember that the deflection had not been suspected until 
Einstein predicted it as a consequence of Relativity. A 
theory which predicts a hitherto unknown phenomenon 
afterwards verified by observation stands on firmer ground 
than a theory invented to account for some known observa- 
tional effect, mainly because there is usually a variety of 
different hypotheses which can be suggested to fit a given 
frame of facts. And whatever modifications future re- 
search may require, nothing can obscure the dramatic 
character of the success which has attended this prediction, 
made by Einstein in 1915, and tested by astronomers at 
the eclipses of 1919 and 1922, and now generally accepted 
as fulfilled. 

General Conclusions. 

The Restricted Theory of Relativity may be regarded as 
complete : future investigations will add nothing to it. If 
we accept the two fundamental axioms on which it is based, 
the conclusions follow as a matter of formal logic. In 
process of time the characteristic ideas of the theory will 
become familiar to the man in the street, their acceptance 
will then become a matter of course, and their revolutionary 
nature will be forgotten. If intercourse should ever become 
practicable between beings in worlds separating with 
velocities comparable with that of light, the theory would 
seriously affect conduct, but this contingency seems so 
remote and improbable that it may safely be discounted. 

The General Theory of Relativity may in a sense be said 
to be still in the making. Certainly its implications are 
far more controversial. To say that Einstein's Law of 
Space-time has superseded Newton's Law of Gravitation 
is not the same as affirming that Einstein's theory fits in 
with all the phenomena of modern physics. No link has, 
for example, yet been forged between the theory of Rela- 


tivity and the Quantum theory. But if we accept the 
principles of Restricted Relativity, Newton's Law of Gravi- 
tation cannot stand in its existing form for the simple 
reason that it is ambiguous. Apart from the doubt in the 

Newtonian expression —$- as to the meaning of mass, 

which arises because the mass changes with the velocity, 
we have seen that the value of r also depends on the 
circumstances of the observer. 

The universality of character which was the most striking 
feature of the law has therefore disappeared. No doubt 
the statement by which the law is expressed could be 
modified so as to remove these difficulties of interpretation. 
It is not, however, worth while attempting to do so, because 
it cannot in any case be expressed in a form which will be 
true for all observers. It is not therefore the kind of law 
which our previous discussions have shown a law of Nature 
should be ; at the same time calculations based on it give 
results which agree closely with those deduced from 
Einstein's general theory. Formally the two theories have 
nothing in common ; they are built upon contrary hypo- 
theses. Newton assumes an absolute space and an 
absolute time, and his laws of motion are bound up with 
these hypotheses. Einstein treats these suppositions as 
untrue, and creates a space-time (not a space and time) 
domain. The great achievement in his theory is the fact 
that he is able independently of axes of reference to identify 
a unique track in space-time uniting two events, the 
generalisation of a straight line in Euclid, and to specify 
this path both in a field devoid of matter and in a field 
influenced by matter. The theory of Relativity, whether 
restricted or general, is a theory of geodesies. The treat- 
ment in this book is confined to those aspects of the subject 
which may reasonably be regarded as established or at least 
supported by strong evidence, although further investiga- 


tions may introduce minor modifications. But the theory 
has led physicists to formulate further hypotheses which at 
present must be considered mainly speculative. What is 
the nature of the structure of the Universe ; is it finite or 
infinite, is it continuous or discrete, is its substance matter 
or events? Some, perhaps all, of these questions may 
never receive a final and complete answer. In regard to 
the first of these inquiries there is, however, a slight balance 
of evidence in favour of supposing that the Universe is 
finite, but of course unbounded, and an estimate of its size 
is expressed by the statement that a ray of light emitted 
from a source would, if unimpeded, travel round the 
Universe and return to its point of departure after a 
thousand million years. It is not inconceivable that science 
may at some future time be able to devise and execute a 
practical test which will decide the question. But this is 
a superficial type of knowledge. The theory of Relativity 
may describe the laws which reality obeys, and trace its 
structure : this indeed is all it sets out to do. Of the inner 
nature of things, it says nothing : this is left to the phil- 
osopher, who says a great deal, but in the end we never 
appear to be much, if any, the better for it. 


1. At what distance would a halfpenny (diameter i inch) 

subtend an angle of i second ? 

2. Taking the mean distance of Mercury from the Sun as 

37,000,000 miles, and the length of Mercury's year as 
88 of our days, show that the speed of Mercury in 
her orbit is slightly less than ^^ lux per second, 
[i lux=o= 186,000 miles.] 

3. It can be deduced from Einstein's space-time law that if 

the speed of a planet is v luxes per second, the major 
axis of its orbit advances i2i> 2 right angles per revolution. 
Using the data of No. 2, show that in 100 of our years 
the major axis of Mercury advances about 43 seconds. 


4. With the data on p. 138, show that — right angles is 

approximately 175 seconds. 

5. Taking the radius of Jupiter as 43,000 miles, and its mass 

as jjfe^ of that of the Sun, show that, according to the 
Einstein theory, the apparent displacement of a star so 
placed that the light -ray to the Earth just touches 
Jupiter's surface is about 0-017 second. 

6. With the data on p. 136 for the spectral shift, prove that 




3. 18-5. 6. ioo seconds. 8. 17 miles. 9. 39,800 stadia. 

1. 4 feet. 2. 2 feet ; 6 inches. 

3. 60 feet ; 3 inches ; 3 inches ; 4. 

4. 2-6 inches; 6 inches. 5. 0-25 ; i=(£) 2 . 

6. Frustrum of pyramid, approx. £ by J by i inch. 

7. 3 inches broad ; f inch thick. 8. Too thin. 


3. (i) Ordinary ; (ii) dial oval ; hands expand and contract 

as they rotate. 

4. As in 3 (ii). 5. 6 3 £° N. of E. 6. No. 

1. 10 yards ; 24 yards. 2. | y feet ; £ z feet. 
3. 9 seconds ; 27 feet. 4. 20 seconds ; 25 seconds. 

5. 32 feet. 6. Bullet ; together ; noise. 


1. (i) P 20, A 16, C 10 ; (ii) P 25, A 20, C 14 seconds past zero. 

2. (i) P 30, A 24, C 1 5 ; (ii) P 37i, A 30, C 21 seconds past zero. 

3. (i) D 12 ahead of A ; (ii) 15. 

4. (i) P 5, A 4, C 10 ; (ii) P 25, A 20, C 26 seconds past zero. 

5. (i) Event at A ; (ii) event at C. 6. Yes. 7. 40 legs. 


1. $ foot when pointing east ; loses 12 minute-spaces per 

hour ; A says the same about O. 

2. A is 1*4 second-spaces ahead of C ; D is 2-8 second-spaces 

ahead of A ; A 24, C 22-6, D 268 seconds past zero. 

3. 8£ seconds ; "j\ luxes. 4. 4 seconds ; same place. 




2. Yes; -£ ; 11-3. 3. Xwm ^ t /=>6 |. 4 

4. 9 luxes ; 12 ; / 5 or -£ lux per second. 

5. 4 seconds, 7 luxes ; -\ second ; A says II. occurs before I. ; 


6. Imaginary ; indeterminate. 7. Real ; fixed. 8. Yes. 

9. III. occurred after I. and II. ; no time-order exists for 

I. and II. 

10. 25 hours ; o. 

I' 8 4- 2. i-6, 3-8, 10-4; (2-5, 5, 14 ; 77-5). 

3. 84 seconds ; parallel to time-axis ; proper time. 
5. 160. 6. 16; 8+ V 28 = 13-29; no. 

!• § , II lux per'second. 2. o, ^- lux per second. 

3. § lux per second perp. toAB; f lux per second along AB 

*• 2 *> 375. 5. ii, 3-17. 


1. Yes ; a parabola ; a straight line. 

2. B says the glass box first stops and then moves with 

upward increasing vertical acceleration. 

3. Yes. 5. 0-000,005. 6. 0-000,000,000,017. 

1. 3-26 miles. 

Wi*3 ■■■■■■' 










Ma agfii 

I Ki£ai5iaF 

.HI He E3 

i ■■■■■■■■■■■■■ 



jraMMm^'fivT JTriffi SAT'S* »tt^^^TC "^^fjr^ 

: *^S ■■■■■■■ HvBvjbvjbvjbvjiiiiI ■■■■■l